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Turn the Beat Around
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For the MTV television film, see Turn the Beat Around (film).
"Turn the Beat Around"
Single by Vicki Sue Robinson
from the album Never Gonna Let You Go
B-side
"Lack of Respect"
Released
April 1976
Format
7" 45 RPM
Genre
Disco
Length
5:35
Label
RCA Victor
Songwriter(s)
Gerald Jackson, Peter Jackson
Vicki Sue Robinson singles chronology
"Baby Now That I Found You"
(1975) "Turn the Beat Around"
(1976) "Never Gonna Let You Go"
(1976)
"Baby Now That I Found You"
(1975) "Turn the Beat Around"
(1976) "Never Gonna Let You Go"
(1976)
"Turn the Beat Around" is a disco song written by Gerald Jackson and Peter Jackson and performed by Vicki Sue Robinson in 1976 (see 1976 in music), originally appearing on her debut album, Never Gonna Let You Go. Released as a single, the song went to #10 on the Billboard pop charts, and #73 on the soul chart.[1] Robinson received a grammy nomination for best female pop vocal. The track went to number one on the disco chart for four weeks.[2] "Turn the Beat Around" is considered a disco classic and is featured on many compilation albums.
Contents
1 Background
2 Chart performance
2.1 Weekly charts
2.2 Year-end charts
3 Laura Branigan version
3.1 Track listings
4 Gloria Estefan version
4.1 Release
4.2 Critical reception
4.3 Certifications
4.4 Chart performance
4.4.1 Weekly charts
4.4.2 Year-end charts
5 Other versions
6 References
7 External links
Background[edit]
"Turn the Beat Around" was written by brothers Gerald and Peter Jackson of the R&B outfit Touch of Class. Peter Jackson knew Al Garrison, an engineer at Associated Studios in New York, via his [i.e. Peter Jackson's] work as a session drummer, and it was at Associated Studios that Touch of Class cut its own demos. Peter Jackson recalls that one Sunday at noontime "I called Al and said...we want to come in and [cut a] demo...He was leaving at four...He said: 'My girl[friend]’s coming to pick me up for dinner. You have to be done [by then]."[3]
Garrison's girlfriend turned out to be singer Vicki Sue Robinson whose debut album was nearing completion requiring one additional track. On arriving at Associated Studios that Sunday, Robinson overheard the playback of the "Turn the Beat Around" demo which Touch of Class had just recorded and according to Peter Jackson said: "Oh, man, I’ve gotta have that song." Gerald and Peter Jackson initially demurred, wishing to submit "Turn the Beat Around" along with four earlier demos to be green-lighted for the Touch of Class debut album.[3]
Peter Jackson - "Monday [the next day], Gerald and I go up to Midland [Touch of Class' label]. We’re excited because we know this song ["Turn the Beat Around"] is slammin’...[Midland] took the other four songs and they passed on that one. They said: 'We don’t like that one. The lyrics move too fast. You have that jungle beat in there. It’s not what’s happening'." Peter Jackson resultantly called Vicki Sue Robinson to give her the song for her album. When Jackson told Robinson: "'I’ll meet you down on Thirty-Fourth Street [with the demo]' she said: 'I [already] made Al give me a copy.'"[3]
Robinson recorded "Turn the Beat Around" on September 26, 1975, cutting her lead vocal in a single take after recording her own multi-tracked chorale vocals. Like the other cuts on Robinson's debut album Never Gonna Let You Go, "Turn the Beat Around" was recorded at RCA Studios with producer Warren Schatz who recalls the basic master of the song was recorded "on a Friday after a very depressing week of rain [and] I hated [the track]! I listened to it in my office and I just couldn't get it. It had been such a bad week that I just couldn't hear anything with an open mind. Then David Todd, the head of disco promotion at RCA, came into my office and he went crazy over the track! He convinced me to finish it as soon as possible."[4]
Issued as a single in February 1976 "Turn the Beat Around" became a club smash subsequently breaking on Top 40 radio in Boston - where it would reach #1 that June - to make a gradual ascent on the national Pop chart: the Billboard Hot 100 to reach a #10 peak in August 1976.
Chart performance[edit]
Weekly charts[edit]
Chart (1976)
Peak
position
Australia Kent Music Report[5] 28
Belgium 17
Canada RPM Top Singles[6] 14
France 44
Netherlands 11
UK Singles Chart 51
US Billboard Hot 100[7] 10
US Billboard Easy Listening 43
US Billboard Dance 1
US Billboard Soul Singles[1] 73
US Cash Box Top 100[8] 15
Year-end charts[edit]
Chart (1976)
Rank
Canada [9] 129
US Billboard Hot 100[10] 38
US Cash Box [11] 67
Laura Branigan version[edit]
"Turn the Beat Around"
Single by Laura Branigan
from the album Laura Branigan
Released
1990 (1990)
Format
12"
Genre
Dance-pop
Hi-NRG
Length
4:22
Label
Atlantic
Songwriter(s)
Gerald Jackson, Peter Jackson
Producer(s)
Laura Branigan
Steve Lindsey
Laura Branigan singles chronology
"Never in a Million Years"
(1990) "Turn the Beat Around"
(1990) "Didn't We Almost Win It All"
(1993)
"Never in a Million Years"
(1990) "Turn the Beat Around"
(1990) "Didn't We Almost Win It All"
(1993)
Laura Branigan covered the song in 1990 becoming the first major artist to do so. It was released as the third and final single from her self-titled sixth studio album, however only to radio and clubs. The song was co-produced by Branigan and Steve Lindsey for the album, with several remix versions following garnering significant play in Hi-NRG clubs.
Track listings[edit]
2 Track 12" single[12]
No.
Title
Length
1. "Turn the Beat Around" (Popcussion mix) 7:42
2. "Turn the Beat Around" (Don & Vic mix) 6:25
Promo 12" single[13]
No.
Title
Length
1. "Turn the Beat Around" (Popcussion mix) 7:42
2. "Turn the Beat Around" (Don & Vic mix) 6:25
3. "Turn the Beat Around" (Club House mix) 6:35
4. "Turn the Beat Around" (The Branipella mix) 3:53
5. "Turn the Beat Around" (Radio edit) 4:05
Gloria Estefan version[edit]
"Turn the Beat Around"
Single by Gloria Estefan
from the album Hold Me, Thrill Me, Kiss Me
Released
September 23, 1994 (1994-09-23)
Format
7"
12"
cassette
CD
Recorded
1993–1994
Genre
Dance-pop
disco
hi-NRG
Length
3:52
Label
Epic
Songwriter(s)
Gerald Jackson
Peter Jackson
Gloria Estefan singles chronology
"Mi Tierra de Tradición"
(1994) "Turn the Beat Around"
(1994) "Hold Me, Thrill Me, Kiss Me"
(1994)
"Mi Tierra de Tradición"
(1994) "Turn the Beat Around"
(1994) "Hold Me, Thrill Me, Kiss Me"
(1994)
Music video
"Turn the Beat Around" on YouTube
In 1994 (see 1994 in music), the song was recorded by Gloria Estefan for the soundtrack to the film The Specialist.
Release[edit]
Released as a single, it became a hit reaching #13 on the Billboard Hot 100 and is also featured on Estefan's fourth solo album Hold Me, Thrill Me, Kiss Me. She also sang the song as her opening performance in VH1's first ever Divas Live. She also took the song to the top spot on Billboard's Hot Dance Club Play chart, making it her first number-one song on that chart in English.
Critical reception[edit]
Music & Media wrote about the song: "Estefan revives her Miami Sound Machine days of fatback disco by covering an old Vicky Sue Robinson song."[14]
Network 40 wrote: "A classic remake of the 1976 Top 10 gem by Vicki Sue Robinson. Uptempo flavor spiced with trademark Miami Sound Machine overtones, this number is a multi-format hit."[15]
Certifications[edit]
Country
Certification
Sales
Australia (ARIA) Gold[16] 35,000
United States Gold 500,000
Chart performance[edit]
Weekly charts[edit]
Chart (1994)
Peak
position
Australia (ARIA Chart)[17] 8
Belgium (Ultratop 50 Flanders) 24
Canada (RPM)[18] 26
Canada Dance (RPM)[19] 7
Europe (Eurochart Hot 100)[20] 66
Germany (Media Control Charts) 51
Netherlands (Dutch Top 40)[21] 33
Netherlands (Single Top 100)[22] 27
New Zealand (RIANZ) 18
United Kingdom (UK Singles Chart) 21
United Kingdom (UK Dance Singles Chart)[23] 38
U.S. Billboard Hot 100 13
U.S. Billboard Hot Adult Contemporary Tracks 4
U.S. Billboard Hot Dance Club Play 1
U.S. Billboard Mainstream Top 40 20
U.S. Cash Box[24] 14
Year-end charts[edit]
Chart (1995)
Rank
U.S. Billboard Hot 100[25] 87
Other versions[edit]
The chorus is used in the lead out of the 1982 Soft Cell song "Memorabilia".[26]
Lil Suzy covered the song for her 1993 album Back to Dance. It was released as the lead single.
Cobra Starship covered the song for the MTV movie of the same name.
A television advertisement for "I Can't Believe It's Not Butter" featuring Megan Mullally contains the parody "Turn the Tub Around"[27]
In the second Star Wars Family Guy episode, the song is parodied as "Turn the Ship Around".
The 2005 song "Perfection" by Dannii Minogue and the Soul Seekerz samples "Turn the Beat Around".
This song was covered many times on American Idol. Both Carmen Rasmusen and Diana DeGarmo performed this song at the Top 6 of American Idol Season 2 and Season 3, respectively. Haley Scarnato also covered this on the sixth season of American Idol. Jessica Sanchez performed the song on the Top 12 of American Idol Season 11.
This song is featured on the 2012 movie Pitch Perfect, as part of the performance by the group Barden Bellas.
The song appeared in the 2015 movie The Martian directed by Ridley Scott and starring Matt Damon.
The song is featured in the 2016 Netflix series "The Get Down" created by Baz Luhrmann.
References[edit]
^ Jump up to: a b Whitburn, Joel (2004). Top R&B/Hip-Hop Singles: 1942-2004. Record Research. p. 497.
Jump up ^ Whitburn, Joel (2004). Hot Dance/Disco: 1974-2003. Record Research. p. 219.
^ Jump up to: a b c "Archived copy". Archived from the original on 2014-03-04. Retrieved 2014-03-01.
Jump up ^ Arena, James (2014). First Legends of Disco. Bloomington IN: Author House. p. 274. ISBN 978-1-4918-4828-9.
Jump up ^ "Forum - 1970 (ARIA Charts: Special Occasion Charts)". Australian-charts.com. Archived from the original on 2016-06-02. Retrieved 2016-12-04.
Jump up ^ Canada, Library and Archives (17 July 2013). "Image : RPM Weekly".
Jump up ^ Joel Whitburn's Top Pop Singles 1955-1990 - ISBN 0-89820-089-X
Jump up ^ "Cash Box Top 100 8/21/76".
Jump up ^ "Top Singles – Volume 26, No. 14 & 15, January 08 1977". RPM. Library and Archives Canada. Archived from the original on June 10, 2016. Retrieved December 15, 2016.
Jump up ^ "Top 100 Hits of 1976/Top 100 Songs of 1976".
Jump up ^ "Cash Box YE Pop Singles - 1976".
Jump up ^ "Laura Branigan - Turn the Beat Around (Vinyl, 12") at Discogs". Discogs. Retrieved December 13, 2013.
Jump up ^ "Laura Branigan - Turn the Beat Around (Vinyl, 12") at Discogs". Discogs. Retrieved December 13, 2013.
Jump up ^ "New Releases: Singles" (PDF). Music & Media (1994-02-12, page 12). Retrieved 2018-02-26.
Jump up ^ "Mainstream: Music Meeting" (PDF). Network 40 (1994-09-09, page 20). Retrieved 2018-02-21.
Jump up ^ Ryan, Gavin (2011). Australia's Music Charts 1988–2010. Mt. Martha, VIC, Australia: Moonlight Publishing.
Jump up ^ "australian-charts.com > Gloria Estefan discography". Hung Medien. Retrieved 2015-09-20.
Jump up ^ "RPM (Nov 21, 1994)". RPM. Retrieved 2018-03-09.
Jump up ^ "RPM Dance (Dec 05, 1994)". RPM. Retrieved 2018-03-08.
Jump up ^ "Eurochart Hot 100" (PDF). Music & Media. Retrieved 2018-03-15.
Jump up ^ "Nederlandse Top 40 – Gloria Estefan" (in Dutch). Dutch Top 40.
Jump up ^ "Dutchcharts.nl – Gloria Estefan – Turn the Beat Around" (in Dutch). Single Top 100.
Jump up ^ "Official UK Dance Singles Chart (09 October 1994-15 October 1994)". officialcharts.com. Retrieved 2018-02-09.
Jump up ^ "Cash Box Top 100 Pop Singles" (PDF). Cash Box (1994-11-26). Retrieved 19 February 2018.
Jump up ^ "Billboard Top 100 - 1995". Longboredsurfer.com. Archived from the original on 2009-08-15. Retrieved 2010-08-27.
Jump up ^ "Memorabilia Lyrics - Soft Cell". Sing365.com. Retrieved 2016-10-11.
Jump up ^ Erika Brooks Adickman (2010-02-01). "Turn The Tub Around: I Can't Believe It's A Commercial". Idolator. Retrieved 2016-10-11.
External links[edit]
Lyrics of this song at MetroLyrics
show
v
t
e
Laura Branigan singles
Branigan
"All Night with Me"
"Gloria"
Branigan 2
"Solitaire"
"How Am I Supposed to Live Without You"
"Deep in the Dark"
Self Control
"Self Control"
"The Lucky One"
"Ti Amo"
"Satisfaction"
Hold Me
"Spanish Eddie"
"Hold Me"
"I Found Someone"
Touch
"Shattered Glass"
"Power of Love"
"Cry Wolf"
Laura Branigan
"Moonlight on Water"
"Never in a Million Years"
"Turn the Beat Around"
Over My Heart
"It's Been Hard Enough Getting Over You"
"How Can I Help You to Say Goodbye"
The Best of Branigan
"Dim All the Lights"
Other singles
"Looking Out for Number One"
"Tell Him"
"Come into My Life"
Other songs
"Mama"
"Foolish Lullaby"
show
v
t
e
Gloria Estefan singles
Eyes of Innocence
"Dr. Beat"
"Prisoner of Love"
"I Need a Man"
Primitive Love
"Conga"
"Bad Boy"
"Words Get in the Way"
"Falling in Love (Uh-Oh)"
Let It Loose/Anything for You
"Rhythm Is Gonna Get You"
"Betcha Say That"
"Can't Stay Away from You"
"Anything for You"
"1-2-3"
Cuts Both Ways
"Don't Wanna Lose You"
"Get on Your Feet"
"Here We Are"
"Oye Mi Canto (Hear My Voice)"
"Cuts Both Ways"
Exitos de Gloria Estefan
"Renacer"
Into the Light
"Coming Out of the Dark"
"Seal Our Fate"
"Remember Me with Love"
"Can't Forget You"
"Nayib’s Song (I Am Here for You)"
"Live for Loving You"
Greatest Hits
"Always Tomorrow"
"Megamix"
"I See Your Smile"
"Go Away"
Mi Tierra
"Mi Tierra"
"Con Los Años Que Me Quedan"
"Tradición"
"Montuno
"¡Sí Señor!...""
"Mi Buen Amor"
"Ayer"
Hold Me, Thrill Me, Kiss Me
"Turn the Beat Around"
"Hold Me, Thrill Me, Kiss Me"
"Everlasting Love"
"It's Too Late"
"Cherchez La Femme"
Abriendo Puertas
"Abriendo Puertas"
"Más Allá"
"Tres Deseos"
"Dulce Amor"
"La Parranda"
"Lejos De Ti"
"Felicidad"
"Abriendo Puertas A La Navidad"
Destiny
"Reach"
"You'll Be Mine (Party Time)"
"I'm Not Giving You Up"
"Higher"
"Show Me the Way Back to Your Heart"
"No Pretendo"
gloria!
"Heaven's What I Feel/Corazón Prohibido"
"Oye!"
"Cuba Libre"
"Don't Let This Moment End"
"Don't Stop"
Alma Caribeña - Caribbean Soul
"No Me Dejes de Querer"
"Como Me Duele Perderte"
"Tres Gotas De Agua Bendita"
"Tengo Que Decirte Algo"
"Me Voy"
"Por Un Beso"
Greatest Hits Vol. II
"You Can't Walk Away from Love"
"Out of Nowhere"
"Y-Tu-Conga"
Unwrapped
"Wrapped/Hoy"
"I Wish You"
"Te Amaré"
"Tu Fotografía"
90 Millas
"No Llores"
"Me Odio"
"Píntame De Colores"
Miss Little Havana
"Wepa"
"Hotel Nacional"
Other notable songs & Collaborations
"En El Jardín"
"Santo Santo"
"Music of My Heart"
"What More Can I Give
"Dr. Pressure"
"Somos El Mundo"
Book
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Nico and the Niners
From Wikipedia, the free encyclopedia
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"Nico and the Niners"
Single by Twenty One Pilots
from the album Trench
A-side
"Jumpsuit"
Released
July 11, 2018[1]
Format
Digital download
Genre
Reggae-rap[2]
psychedelic[2]
Length
3:47
Label
Fueled by Ramen
Songwriter(s)
Tyler Joseph, Josh Dun
Twenty One Pilots singles chronology
"Jumpsuit"
(2018) "Nico and the Niners"
(2018) "Levitate"
(2018)
Music video
"Nico and the Niners" on YouTube
"Nico and the Niners" is a song written and recorded by American musical duo Twenty One Pilots. It was released on July 11, 2018 as one of the singles from their upcoming fifth studio album Trench (2018), alongside "Jumpsuit".[3][4] The song peaked at No. 79 on the US Billboard Hot 100 chart.[5]
Contents
1 Composition
2 Music video
3 Commercial performance
4 Charts
5 References
Composition[edit]
"Nico and the Niners" is a reggae, rap, and psychedelic song with a length of three minutes and forty-seven seconds.[2] It features a rap verse, some vocal audio clips played backwards and a section with a reversed drum beat. Additionally, there are sections using a vocoder effect on clips of the vocals.[6] The song is played in a A minor key, and 140 beats per minute.[7]
Music video[edit]
A music video for "Nico and the Niners" was released on July 26, 2018.[8] The video shows singer Tyler Joseph packing a backpack and meeting drummer Josh Dun in a faded city (known as Dema) square, where they shake hands before joining a group of torch-wielding rebels and perform the song. The video veers between that mysterious thread and the mystical activities of mysterious red-hooded robed figures.
The video was filmed in Kharkiv and Kyiv, Ukraine.
As of August 27, 2018, the music video has received over 14 million views on YouTube.
Commercial performance[edit]
"Nico and the Niners" made its debut on the Bubbling Under Hot 100 Singles chart at No. 6; the following week, it peaked at No. 79 on the Billboard Hot 100. The song also peaked at No. 7 on the Hot Rock Songs chart.
Charts[edit]
Chart (2018)
Peak
position
Czech Republic (Singles Digitál Top 100)[9]
2
France (SNEP)[10]
158
Greece International Digital Singles (IFPI)[11]
46
Hungary (Single Top 40)[12]
24
Hungary (Stream Top 40)[13]
36
New Zealand Hot Singles (RMNZ)[14]
7
UK Singles (Official Charts Company)[15]
88
US Billboard Hot 100[16]
79
US Hot Rock Songs (Billboard)[17]
7
References[edit]
Jump up ^ "Jumpsuit / Nico And The Niners - Single". iTunes. July 11, 2018. Retrieved July 20, 2018.
^ Jump up to: a b c Reed, Ryan (July 11, 2018). "Hear Twenty One Pilots' New Songs". Rolling Stone. Retrieved July 20, 2018.
Jump up ^ "Top 40 Future/M Releases". AllAccess. Archived from the original on July 20, 2018.
Jump up ^ "Jumpsuit / Nico And The Niners - Single on Spotify". Spotify. July 11, 2018. Retrieved July 20, 2018.
Jump up ^ "Twenty One Pilots Storm Back with 'Jumpsuit' and 'Nico and the Niners': Listen". Billboard. Retrieved 2018-07-20.
Jump up ^ "Jumpsuit / Nico And The Niners - Single on Spotify". Spotify. July 11, 2018. Retrieved July 20, 2018.
Jump up ^ "Key & BPM for Nico and the Niners by twenty one pilots". TuneBat. July 11, 2018. Retrieved July 20, 2018.
Jump up ^ Kaufman, Gil (26 July 2018). "Twenty One Pilots Hint at Rebel Uprising in Cryptic 'Nico and the Niners' Video: Watch". Billboard. Retrieved 26 July 2018.
Jump up ^ "ČNS IFPI" (in Czech). Hitparáda – Digital Top 100 Oficiální. IFPI Czech Republic. Note: insert 201829 into search. Retrieved July 23, 2018.
Jump up ^ "Lescharts.com – {{{artist}}} – {{{song}}}" (in French). Les classement single. Retrieved July 23, 2018.
Jump up ^ "Official IFPI Charts – Digital Singles Chart (International) – Week: 28/2018". IFPI Greece. Archived from the original on July 26, 2018. Retrieved July 26, 2018.
Jump up ^ "Archívum – Slágerlisták – MAHASZ" (in Hungarian). Single (track) Top 40 lista. Magyar Hanglemezkiadók Szövetsége. Retrieved July 21, 2018.
Jump up ^ "Archívum – Slágerlisták – MAHASZ" (in Hungarian). Stream Top 40 slágerlista. Magyar Hanglemezkiadók Szövetsége. Retrieved July 30, 2018.
Jump up ^ "NZ Hot Singles Chart". Recorded Music NZ. July 23, 2018. Retrieved July 20, 2018.
Jump up ^ "Official Singles Chart Top 100". Official Charts Company. Retrieved July 20, 2018.
Jump up ^ "Twenty One Pilots Chart History (Hot 100)". Billboard. Retrieved July 24, 2018.
Jump up ^ "Twenty One Pilots Chart History (Hot Rock Songs)". Billboard. Retrieved July 24, 2018.
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text:
The Princess and the Frog (soundtrack)
From Wikipedia, the free encyclopedia
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The Princess and the Frog: Original Songs and Score
Soundtrack album by Various Artists
Released
November 23, 2009
Recorded
2009
Genre
Jazz, soul, gospel, R&B, zydeco
Label
Walt Disney
Producer
Randy Newman
Walt Disney Animation Studios chronology
Bolt
(2008)Bolt2008 The Princess and the Frog
(2009) Tangled
(2010)Tangled2010
Randy Newman chronology
Leatherheads
(2008) Leatherheads2008 The Princess and the Frog
(2009) The Princess and the Frog2009 Toy Story 3
(2010) Toy Story 32010
Singles from The Princess and the Frog: Original Songs and Score
"Never Knew I Needed"
Released: October 27, 2009
The Princess and the Frog: Original Songs and Score is the soundtrack of the 2009 Disney animated film The Princess and the Frog. It was released by Walt Disney Records on November 23, 2009, just a day before the limited release of the film in New York City and Los Angeles. It contains ten original songs and seven score pieces, all but one of which were composed, arranged and conducted by composer Randy Newman.[1] "Never Knew I Needed" was written and performed by Ne-Yo. The song had an accompanying music video which featured rotation on Disney Channel. The song was also sent to rhythmic radio on October 27, 2009.[2] The songs are performed by various artists most of which lend their voices to characters in the film. The score features African-American-influenced styles including jazz, zydeco, blues and gospel.[3]
"Almost There" and "Down in New Orleans" were both nominated in the Best Original Song category at the 82nd Academy Awards; they lost to "The Weary Kind". "Down in New Orleans" was also nominated for the Grammy Award for Best Song Written for a Motion Picture, Television or Other Visual Media at the 2011 Grammy Awards.[4]
Contents
[hide]
1 Track listing
2 Charts
3 Reception
4 References
5 External links
Track listing[edit]
All tracks written by Randy Newman except "Never Knew I Needed" which was written by Ne-Yo.
No.
Title
Performer
Length
1. "Never Knew I Needed" Ne-Yo 3:38
2. "Down in New Orleans (Prologue)" Anika Noni Rose 0:28
3. "Down in New Orleans" Dr. John 2:27
4. "Almost There" Anika Noni Rose 2:24
5. "Friends on the Other Side" Keith David 3:35
6. "When We're Human" Michael-Leon Wooley, Bruno Campos and Anika Noni Rose featuring Terence Blanchard 2:22
7. "Gonna Take You There" Jim Cummings featuring Terrance Simien on Accordion 1:46
8. "Ma Belle Evangeline" Jim Cummings featuring Terence Blanchard 1:56
9. "Dig a Little Deeper" Jenifer Lewis featuring the Pinnacle Gospel Choir and Anika Noni Rose 2:48
10. "Down in New Orleans (Finale)" Anika Noni Rose 1:38
11. "Fairy Tale/Going Home" (score) Randy Newman 4:17
12. "I Know this Story" (score) Randy Newman 5:27
13. "The Frog Hunters/Gator Down" (score) Randy Newman 6:04
14. "Tiana's Bad Dream" (score) Randy Newman 6:22
15. "Ray Laid Low" (score; with a reprise of "Almost There") Randy Newman; cameo appearance by Anika Noni Rose 3:22
16. "Ray/Mama Odie" (score) Randy Newman 4:01
17. "This is Gonna Be Good" (score) Randy Newman 3:20
Charts[edit]
Chart
Peak
position
U.S. Billboard 200[5] 80
Reception[edit]
Professional ratings
Review scores
Source
Rating
Allmusic [6]
Filmtracks [7]
In 2010, Rhapsody called it one of the all-time great Disney & Pixar soundtracks.[8]
References[edit]
Jump up ^ "Disney's The Princess and the Frog soundtrack". Starpulse. Retrieved December 19, 2009.
Jump up ^ FMQB
Jump up ^ "The Princess and the Frog Soundtrack". Archived from the original on December 13, 2009. Retrieved December 19, 2009.
Jump up ^ Morris, Christopher (December 1, 2010). "53rd Annual Grammy nominees". Variety. Retrieved December 1, 2010.
Jump up ^ "The Princess and the Frog [Original Songs and Score] – Randy Newman". Billboard. Retrieved 2012-03-06.
Jump up ^ Allmusic review
Jump up ^ [1]
Jump up ^ 10 Essential Disney & Pixar Soundtracks Archived October 2, 2011, at the Wayback Machine. Referenced July 27, 2010
External links[edit]
Disney Records website
show
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Randy Newman
12 Songs
Sail Away
Good Old Boys
Little Criminals
Born Again
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Land of Dreams
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Dark Matter
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Randy Newman Live
Compilations
Lonely at the Top: The Best of Randy Newman
Guilty: 30 Years of Randy Newman
The Randy Newman Songbook Vol. 1
The Randy Newman Songbook Vol. 2
Musicals
Randy Newman's Faust
Soundtracks
Toy Story
James and the Giant Peach
Avalon
Meet the Parents
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Songs
"Mama Told Me Not to Come"
"Sail Away"
"He Gives Us All His Love"
"Simon Smith and the Amazing Dancing Bear"
"Political Science"
"You Can Leave Your Hat On"
"Louisiana 1927"
"Rednecks"
"Short People"
"I Love L.A."
"You've Got a Friend in Me"
"When She Loved Me"
"We Belong Together"
"It's a Jungle Out There"
Related
Discography
Nilsson Sings Newman
Alfred Newman
David Newman
Emil Newman
Joey Newman
Lionel Newman
Thomas Newman
show
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Soundtrack
"Never Knew I Needed"
"Down in New Orleans"
"Almost There"
"Friends on the Other Side"
"When We're Human"
Other
Tiana
"Greenbacks" (Once Upon a Time)
Retrieved from "https://en.wikipedia.org/w/index.php?title=The_Princess_and_the_Frog_(soundtrack)&oldid=847320134"
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text:
Invertible matrix
From Wikipedia, the free encyclopedia
Jump to: navigation, search
In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that
A B = B A = I n {\displaystyle \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n}\ }
where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A and is called the inverse of A, denoted by A−1.
A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that a square matrix randomly selected from a continuous uniform distribution on its entries will almost never be singular.
Non-square matrices (m-by-n matrices for which m ≠ n) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n, then A has a left inverse: an n-by-m matrix B such that BA = In. If A has rank m, then it has a right inverse: an n-by-m matrix B such that AB = Im.
Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.
While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any ring. However, in the case of the ring being commutative, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated since a notion of rank does not exist over rings.
The set of n × n invertible matrices together with the operation of matrix multiplication form a group, the general linear group of degree n.
Contents
[hide]
1 Properties
1.1 The invertible matrix theorem
1.2 Other properties
1.3 In relation to its adjugate
1.4 In relation to the identity matrix
1.5 Density
2 Examples
3 Methods of matrix inversion
3.1 Gaussian elimination
3.2 Newton's method
3.3 Cayley–Hamilton method
3.4 Eigendecomposition
3.5 Cholesky decomposition
3.6 Analytic solution
3.6.1 Inversion of 2 × 2 matrices
3.6.2 Inversion of 3 × 3 matrices
3.6.3 Inversion of 4 × 4 matrices
3.7 Blockwise inversion
3.8 By Neumann series
3.9 P-adic approximation
4 Derivative of the matrix inverse
5 Generalized inverse
6 Applications
6.1 Regression/least squares
6.2 Matrix inverses in real-time simulations
6.3 Matrix inverses in MIMO wireless communication
7 See also
8 Notes
9 References
10 Further reading
11 External links
Properties[edit]
The invertible matrix theorem[edit]
Let A be a square n by n matrix over a field K (for example the field R of real numbers). The following statements are equivalent, i.e., for any given matrix they are either all true or all false:
A is invertible, i.e. A has an inverse, is nonsingular, or is nondegenerate.
A is row-equivalent to the n-by-n identity matrix In.
A is column-equivalent to the n-by-n identity matrix In.
A has n pivot positions.
det A ≠ 0. In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring.
A has full rank; that is, rank A = n.
The equation Ax = 0 has only the trivial solution x = 0
Null A = {0}
The equation Ax = b has exactly one solution for each b in Kn.
The columns of A are linearly independent.
The columns of A span Kn
Col A = Kn
The columns of A form a basis of Kn.
The linear transformation mapping x to Ax is a bijection from Kn to Kn.
There is an n-by-n matrix B such that AB = In = BA.
The transpose AT is an invertible matrix (hence rows of A are linearly independent, span Kn, and form a basis of Kn).
The number 0 is not an eigenvalue of A.
The matrix A can be expressed as a finite product of elementary matrices.
The matrix A has a left inverse (i.e. there exists a B such that BA = I) or a right inverse (i.e. there exists a C such that AC = I), in which case both left and right inverses exist and B = C = A−1.
Other properties[edit]
Furthermore, the following properties hold for an invertible matrix A:
(A−1)−1 = A;
(kA)−1 = k−1A−1 for nonzero scalar k;
(Ax)+ = x+A−1 where + denotes the Moore–Penrose inverse and x is a vector;
(AT)−1 = (A−1)T;
For any invertible n-by-n matrices A and B, (AB)−1 = B−1A−1. More generally, if A1, ..., Ak are invertible n-by-n matrices, then (A1A2⋅⋅⋅Ak−1Ak)−1 = A−1
kA−1
k−1⋯A−1
2A−1
1;
det A−1 = (det A)−1.
A matrix that is its own inverse, i.e. such that A = A−1 and A2 = I, is called an involutory matrix.
In relation to its adjugate[edit]
The adjugate of a matrix A {\displaystyle A} can be used to find the inverse of A {\displaystyle A} as follows:
If A {\displaystyle A} is an n × n {\displaystyle n\times n} invertible matrix, then
A − 1 = 1 det ( A ) adj  ( A ) {\displaystyle A^{-1}={\frac {1}{\det(A)}}\operatorname {adj} (A)} .
In relation to the identity matrix[edit]
It follows from the theory of matrices that if
A B = I {\displaystyle \mathbf {AB} =\mathbf {I} \ }
for finite square matrices A and B, then also
B A = I {\displaystyle \mathbf {BA} =\mathbf {I} \ } [1]
Density[edit]
Over the field of real numbers, the set of singular n-by-n matrices, considered as a subset of Rn×n, is a null set, i.e., has Lebesgue measure zero. This is true because singular matrices are the roots of the polynomial function in the entries of the matrix given by the determinant. Thus in the language of measure theory, almost all n-by-n matrices are invertible.
Furthermore, the n-by-n invertible matrices are a dense open set in the topological space of all n-by-n matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of n-by-n matrices.
In practice however, one may encounter non-invertible matrices. And in numerical calculations, matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be ill-conditioned.
Examples[edit]
Consider the following 2-by-2 matrix:
A = ( − 1 3 2 1 − 1 ) . {\displaystyle \mathbf {A} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\1&-1\end{pmatrix}}.}
The matrix A {\displaystyle \mathbf {A} } is invertible. To check this, one can compute that det A = − 1 / 2 {\displaystyle \det \mathbf {A} =-1/2} , which is non-zero.
As an example of a non-invertible, or singular, matrix, consider the matrix
B = ( − 1 3 2 2 3 − 1 ) . {\displaystyle \mathbf {B} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\{\tfrac {2}{3}}&-1\end{pmatrix}}.}
The determinant of B {\displaystyle \mathbf {B} } is 0, which is a necessary and sufficient condition for a matrix to be non-invertible.
Methods of matrix inversion[edit]
Gaussian elimination[edit]
Gauss-Jordan elimination is an algorithm that can be used to determine whether a given matrix is invertible and to find the inverse. An alternative is the LU decomposition which generates upper and lower triangular matrices which are easier to invert.
Newton's method[edit]
A generalization of Newton's method as used for a multiplicative inverse algorithm may be convenient, if it is convenient to find a suitable starting seed:
X k + 1 = 2 X k − X k A X k . {\displaystyle X_{k+1}=2X_{k}-X_{k}AX_{k}.}
Victor Pan and John Reif have done work that includes ways of generating a starting seed.[2][3] Byte magazine summarised one of their approaches.[4]
Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix, e.g. the pair of sequences of inverse matrices used in obtaining matrix square roots by Denman-Beavers iteration; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors due to imperfect computer arithmetic.
Cayley–Hamilton method[edit]
The Cayley–Hamilton theorem allows the inverse of A to be expressed in terms of det(A), traces and powers of A [5]
A − 1 = 1 det ( A ) ∑ s = 0 n − 1 A s ∑ k 1 , k 2 , … , k n − 1 ∏ l = 1 n − 1 ( − 1 ) k l + 1 l k l k l ! t r ( A l ) k l , {\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\sum _{s=0}^{n-1}\mathbf {A} ^{s}\sum _{k_{1},k_{2},\ldots ,k_{n-1}}\prod _{l=1}^{n-1}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\mathrm {tr} (\mathbf {A} ^{l})^{k_{l}},}
where n is dimension of A, and tr(A) is the trace of matrix A given by the sum of the main diagonal. The sum is taken over s and the sets of all kl ≥ 0 satisfying the linear Diophantine equation
s + ∑ l = 1 n − 1 l k l = n − 1. {\displaystyle s+\sum _{l=1}^{n-1}lk_{l}=n-1.}
The formula can be rewritten in terms of complete Bell polynomials of arguments tl = - (l - 1)! tr(Al) as
A − 1 = 1 det ( A ) ∑ s = 1 n A s − 1 ( − 1 ) n − 1 ( n − s ) ! B n − s ( t 1 , t 2 , … , t n − s ) . {\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\sum _{s=1}^{n}\mathbf {A} ^{s-1}{\frac {(-1)^{n-1}}{(n-s)!}}B_{n-s}(t_{1},t_{2},\ldots ,t_{n-s}).}
Eigendecomposition[edit]
Main article: Eigendecomposition of a matrix
If matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is invertible and its inverse is given by
A − 1 = Q Λ − 1 Q − 1 {\displaystyle \mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1}}
where Q is the square (N×N) matrix whose ith column is the eigenvector q i {\displaystyle q_{i}} of A and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e., Λ i i = λ i {\displaystyle \Lambda _{ii}=\lambda _{i}} . Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate:
[ Λ − 1 ] i i = 1 λ i {\displaystyle \left[\Lambda ^{-1}\right]_{ii}={\frac {1}{\lambda _{i}}}}
Cholesky decomposition[edit]
Main article: Cholesky decomposition
If matrix A is positive definite, then its inverse can be obtained as
A − 1 = ( L ∗ ) − 1 L − 1 , {\displaystyle \mathbf {A} ^{-1}=(\mathbf {L} ^{*})^{-1}\mathbf {L} ^{-1},}
where L is the lower triangular Cholesky decomposition of A, and L* denotes the conjugate transpose of L.
Analytic solution[edit]
Main article: Cramer's rule
Writing the transpose of the matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices, but this recursive method is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors:
A − 1 = 1 | A | C T = 1 | A | ( C 11 C 21 ⋯ C n 1 C 12 C 22 ⋯ C n 2 ⋮ ⋮ ⋱ ⋮ C 1 n C 2 n ⋯ C n n ) {\displaystyle \mathbf {A} ^{-1}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\mathbf {C} ^{\mathrm {T} }={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}{\begin{pmatrix}\mathbf {C} _{11}&\mathbf {C} _{21}&\cdots &\mathbf {C} _{n1}\\\mathbf {C} _{12}&\mathbf {C} _{22}&\cdots &\mathbf {C} _{n2}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {C} _{1n}&\mathbf {C} _{2n}&\cdots &\mathbf {C} _{nn}\\\end{pmatrix}}}
so that
( A − 1 ) i j = 1 | A | ( C T ) i j = 1 | A | ( C j i ) {\displaystyle \left(\mathbf {A} ^{-1}\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} ^{\mathrm {T} }\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} _{ji}\right)}
where |A| is the determinant of A, C is the matrix of cofactors, and CT represents the matrix transpose.
Inversion of 2 × 2 matrices[edit]
The cofactor equation listed above yields the following result for 2 × 2 matrices. Inversion of these matrices can be done as follows:[6]
A − 1 = [ a b c d ] − 1 = 1 det A [ d − b − c a ] = 1 a d − b c [ d − b − c a ] . {\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}^{-1}={\frac {1}{\det \mathbf {A} }}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}={\frac {1}{ad-bc}}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}.}
This is possible because 1/(ad − bc) is the reciprocal of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes.
The Cayley–Hamilton method gives
A − 1 = 1 det A [ ( tr  A ) I − A ] . {\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det \mathbf {A} }}\left[\left(\operatorname {tr} \mathbf {A} \right)\mathbf {I} -\mathbf {A} \right].}
Inversion of 3 × 3 matrices[edit]
A computationally efficient 3 × 3 matrix inversion is given by
A − 1 = [ a b c d e f g h i ] − 1 = 1 det ( A ) [ A B C D E F G H I ] T = 1 det ( A ) [ A D G B E H C F I ] {\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}^{-1}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,A&\,B&\,C\\\,D&\,E&\,F\\\,G&\,H&\,I\\\end{bmatrix}}^{\mathrm {T} }={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,A&\,D&\,G\\\,B&\,E&\,H\\\,C&\,F&\,I\\\end{bmatrix}}}
(where the scalar A is not to be confused with the matrix A). If the determinant is non-zero, the matrix is invertible, with the elements of the intermediary matrix on the right side above given by
A = ( e i − f h ) D = − ( b i − c h ) G = ( b f − c e ) B = − ( d i − f g ) E = ( a i − c g ) H = − ( a f − c d ) C = ( d h − e g ) F = − ( a h − b g ) I = ( a e − b d ) {\displaystyle {\begin{matrix}A=(ei-fh)&D=-(bi-ch)&G=(bf-ce)\\B=-(di-fg)&E=(ai-cg)&H=-(af-cd)\\C=(dh-eg)&F=-(ah-bg)&I=(ae-bd)\\\end{matrix}}}
The determinant of A can be computed by applying the rule of Sarrus as follows:
det ( A ) = a A + b B + c C . {\displaystyle \det(\mathbf {A} )=aA+bB+cC.}
The Cayley–Hamilton decomposition gives
A − 1 = 1 det ( A ) [ 1 2 ( ( tr  A ) 2 − tr  A 2 ) I − A tr  A + A 2 ] . {\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\left[{\frac {1}{2}}\left((\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} \mathbf {A} ^{2}\right)\mathbf {I} -\mathbf {A} \operatorname {tr} \mathbf {A} +\mathbf {A} ^{2}\right].}
The general 3 × 3 inverse can be expressed concisely in terms of the cross product and triple product. If a matrix A = [ x 0 , x 1 , x 2 ] {\displaystyle \mathbf {A} =\left[\mathbf {x} _{0},\;\mathbf {x} _{1},\;\mathbf {x} _{2}\right]} (consisting of three column vectors, x 0 {\displaystyle \mathbf {x} _{0}} , x 1 {\displaystyle \mathbf {x} _{1}} , and x 2 {\displaystyle \mathbf {x} _{2}} ) is invertible, its inverse is given by
A − 1 = 1 det ( A ) [ ( x 1 × x 2 ) T ( x 2 × x 0 ) T ( x 0 × x 1 ) T ] . {\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}{(\mathbf {x_{1}} \times \mathbf {x_{2}} )}^{\mathrm {T} }\\{(\mathbf {x_{2}} \times \mathbf {x_{0}} )}^{\mathrm {T} }\\{(\mathbf {x_{0}} \times \mathbf {x_{1}} )}^{\mathrm {T} }\\\end{bmatrix}}.}
Note that det ( A ) {\displaystyle \det(A)} is equal to the triple product of x 0 {\displaystyle \mathbf {x_{0}} } , x 1 {\displaystyle \mathbf {x_{1}} } , and x 2 {\displaystyle \mathbf {x_{2}} } —the volume of the parallelepiped formed by the rows or columns:
det ( A ) = x 0 ⋅ ( x 1 × x 2 ) . {\displaystyle \det(\mathbf {A} )=\mathbf {x} _{0}\cdot (\mathbf {x} _{1}\times \mathbf {x} _{2}).}
The correctness of the formula can be checked by using cross- and triple-product properties and by noting that for groups, left and right inverses always coincide. Intuitively, because of the cross products, each row of A − 1 {\displaystyle \mathbf {A} ^{-1}} is orthogonal to the non-corresponding two columns of A {\displaystyle \mathbf {A} } (causing the off-diagonal terms of I = A − 1 A {\displaystyle \mathbf {I} =\mathbf {A} ^{-1}\mathbf {A} } be zero). Dividing by
det ( A ) = x 0 ⋅ ( x 1 × x 2 ) {\displaystyle \det(\mathbf {A} )=\mathbf {x} _{0}\cdot (\mathbf {x} _{1}\times \mathbf {x} _{2})}
causes the diagonal elements of I = A − 1 A {\displaystyle \mathbf {I} =\mathbf {A} ^{-1}\mathbf {A} } to be unity. For example, the first diagonal is:
1 = 1 x 0 ⋅ ( x 1 × x 2 ) x 0 ⋅ ( x 1 × x 2 ) . {\displaystyle 1={\frac {1}{\mathbf {x_{0}} \cdot (\mathbf {x} _{1}\times \mathbf {x} _{2})}}\mathbf {x_{0}} \cdot (\mathbf {x} _{1}\times \mathbf {x} _{2}).}
Inversion of 4 × 4 matrices[edit]
With increasing dimension, expressions for the inverse of A get complicated. For n = 4, the Cayley–Hamilton method leads to an expression that is still tractable:
A − 1 = 1 det ( A ) [ 1 6 ( ( tr  A ) 3 − 3 tr  A tr  A 2 + 2 tr  A 3 ) I − 1 2 A ( ( tr  A ) 2 − tr  A 2 ) + A 2 tr  A − A 3 ] . {\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\left[{\frac {1}{6}}\left((\operatorname {tr} \mathbf {A} )^{3}-3\operatorname {tr} \mathbf {A} \operatorname {tr} \mathbf {A} ^{2}+2\operatorname {tr} \mathbf {A} ^{3}\right)\mathbf {I} -{\frac {1}{2}}\mathbf {A} \left((\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} \mathbf {A} ^{2}\right)+\mathbf {A} ^{2}\operatorname {tr} \mathbf {A} -\mathbf {A} ^{3}\right].}
Blockwise inversion[edit]
Matrices can also be inverted blockwise by using the following analytic inversion formula:
[ A B C D ] − 1 = [ A − 1 + A − 1 B ( D − C A − 1 B ) − 1 C A − 1 − A − 1 B ( D − C A − 1 B ) − 1 − ( D − C A − 1 B ) − 1 C A − 1 ( D − C A − 1 B ) − 1 ] , {\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} (\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\mathbf {CA} ^{-1}&-\mathbf {A} ^{-1}\mathbf {B} (\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\\-(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\mathbf {CA} ^{-1}&(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\end{bmatrix}},}
( 1 ) {\displaystyle (1)\,}
where A, B, C and D are matrix sub-blocks of arbitrary size. (A must be square, so that it can be inverted. Furthermore, A and D − CA−1B must be nonsingular.[7]) This strategy is particularly advantageous if A is diagonal and D − CA−1B (the Schur complement of A) is a small matrix, since they are the only matrices requiring inversion.
This technique was reinvented several times and is due to Hans Boltz (1923),[citation needed] who used it for the inversion of geodetic matrices, and Tadeusz Banachiewicz (1937), who generalized it and proved its correctness.
The nullity theorem says that the nullity of A equals the nullity of the sub-block in the lower right of the inverse matrix, and that the nullity of B equals the nullity of the sub-block in the upper right of the inverse matrix.
The inversion procedure that led to Equation (1) performed matrix block operations that operated on C and D first. Instead, if A and B are operated on first, and provided D and A − BD−1C are nonsingular,[8] the result is
[ A B C D ] − 1 = [ ( A − B D − 1 C ) − 1 − ( A − B D − 1 C ) − 1 B D − 1 − D − 1 C ( A − B D − 1 C ) − 1 D − 1 + D − 1 C ( A − B D − 1 C ) − 1 B D − 1 ] . {\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}&-(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}\mathbf {BD} ^{-1}\\-\mathbf {D} ^{-1}\mathbf {C} (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}&\quad \mathbf {D} ^{-1}+\mathbf {D} ^{-1}\mathbf {C} (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}\mathbf {BD} ^{-1}\end{bmatrix}}.}
( 2 ) {\displaystyle (2)\,}
Equating Equations (1) and (2) leads to
( A − B D − 1 C ) − 1 = A − 1 + A − 1 B ( D − C A − 1 B ) − 1 C A − 1 {\displaystyle (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}=\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} (\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\mathbf {CA} ^{-1}\,}
( 3 ) {\displaystyle (3)\,}
( A − B D − 1 C ) − 1 B D − 1 = A − 1 B ( D − C A − 1 B ) − 1 {\displaystyle (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}\mathbf {BD} ^{-1}=\mathbf {A} ^{-1}\mathbf {B} (\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\,}
D − 1 C ( A − B D − 1 C ) − 1 = ( D − C A − 1 B ) − 1 C A − 1 {\displaystyle \mathbf {D} ^{-1}\mathbf {C} (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}=(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\mathbf {CA} ^{-1}\,}
D − 1 + D − 1 C ( A − B D − 1 C ) − 1 B D − 1 = ( D − C A − 1 B ) − 1 {\displaystyle \mathbf {D} ^{-1}+\mathbf {D} ^{-1}\mathbf {C} (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}\mathbf {BD} ^{-1}=(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\,}
where Equation (3) is the Woodbury matrix identity, which is equivalent to the binomial inverse theorem.
Since a blockwise inversion of an n × n matrix requires inversion of two half-sized matrices and 6 multiplications between two half-sized matrices, it can be shown that a divide and conquer algorithm that uses blockwise inversion to invert a matrix runs with the same time complexity as the matrix multiplication algorithm that is used internally.[9] There exist matrix multiplication algorithms with a complexity of O(n2.3727) operations, while the best proven lower bound is Ω(n2 log n).[10]
By Neumann series[edit]
If a matrix A has the property that
lim n → ∞ ( I − A ) n = 0 {\displaystyle \lim _{n\to \infty }(\mathbf {I} -\mathbf {A} )^{n}=0}
then A is nonsingular and its inverse may be expressed by a Neumann series:[11]
A − 1 = ∑ n = 0 ∞ ( I − A ) n . {\displaystyle \mathbf {A} ^{-1}=\sum _{n=0}^{\infty }(\mathbf {I} -\mathbf {A} )^{n}.}
Truncating the sum results in an "approximate" inverse which may be useful as a preconditioner. Note that a truncated series can be accelerated exponentially by noting that the Neumann series is a geometric sum. As such, it satisfies
∑ n = 0 2 L − 1 ( I − A ) n = ∏ l = 0 L − 1 ( I + ( I − A ) 2 l ) {\displaystyle \sum _{n=0}^{2^{L}-1}(\mathbf {I} -\mathbf {A} )^{n}=\prod _{l=0}^{L-1}(\mathbf {I} +(\mathbf {I} -\mathbf {A} )^{2^{l}})} .
Therefore, only 2 L − 2 {\displaystyle 2L-2} matrix multiplications are needed to compute 2 L {\displaystyle 2^{L}} terms of the sum.
More generally, if A is "near" the invertible matrix X in the sense that
lim n → ∞ ( I − X − 1 A ) n = 0 o r lim n → ∞ ( I − A X − 1 ) n = 0 {\displaystyle \lim _{n\to \infty }(\mathbf {I} -\mathbf {X} ^{-1}\mathbf {A} )^{n}=0\mathrm {~~or~~} \lim _{n\to \infty }(\mathbf {I} -\mathbf {A} \mathbf {X} ^{-1})^{n}=0}
then A is nonsingular and its inverse is
A − 1 = ∑ n = 0 ∞ ( X − 1 ( X − A ) ) n X − 1 . {\displaystyle \mathbf {A} ^{-1}=\sum _{n=0}^{\infty }\left(\mathbf {X} ^{-1}(\mathbf {X} -\mathbf {A} )\right)^{n}\mathbf {X} ^{-1}~.}
If it is also the case that A − X has rank 1 then this simplifies to
A − 1 = X − 1 − X − 1 ( A − X ) X − 1 1 + tr  ( X − 1 ( A − X ) ) . {\displaystyle \mathbf {A} ^{-1}=\mathbf {X} ^{-1}-{\frac {\mathbf {X} ^{-1}(\mathbf {A} -\mathbf {X} )\mathbf {X} ^{-1}}{1+\operatorname {tr} (\mathbf {X} ^{-1}(\mathbf {A} -\mathbf {X} ))}}~.}
P-adic approximation[edit]
This section needs expansion. You can help by adding to it. (February 2015)
If A is a matrix with integer or rational coefficients and we seek a solution in arbitrary-precision rationals, then a p-adic approximation method converges to an exact solution in O ( n 4 log 2  n ) {\displaystyle O(n^{4}\log ^{2}n)} , assuming standard O ( n 3 ) {\displaystyle O(n^{3})} matrix multiplication is used.[12] The method relies on solving n linear systems via Dixon's method of p-adic approximation (each in O ( n 3 log 2  n ) {\displaystyle O(n^{3}\log ^{2}n)} ) and is available as such in software specialized in arbitrary-precision matrix operations, e.g. in IML.[13]
Derivative of the matrix inverse[edit]
Suppose that the invertible matrix A depends on a parameter t. Then the derivative of the inverse of A with respect to t is given by
d A − 1 d t = − A − 1 d A d t A − 1 . {\displaystyle {\frac {\mathrm {d} \mathbf {A} ^{-1}}{\mathrm {d} t}}=-\mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{-1}.}
To derive the above expression for the derivative of the inverse of A, one can differentiate the definition of the matrix inverse A − 1 A = I {\displaystyle \mathbf {A} ^{-1}\mathbf {A} =\mathbf {I} } and then solve for the inverse of A:
d A − 1 A d t = d A − 1 d t A + A − 1 d A d t = d I d t = 0 . {\displaystyle {\frac {\mathrm {d} \mathbf {A} ^{-1}\mathbf {A} }{\mathrm {d} t}}={\frac {\mathrm {d} \mathbf {A} ^{-1}}{\mathrm {d} t}}\mathbf {A} +\mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}={\frac {\mathrm {d} \mathbf {I} }{\mathrm {d} t}}=\mathbf {0} .}
Subtracting A − 1 d A d t {\displaystyle \mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}} from both sides of the above and multiplying on the right by A − 1 {\displaystyle \mathbf {A} ^{-1}} gives the correct expression for the derivative of the inverse:
d A − 1 d t = − A − 1 d A d t A − 1 . {\displaystyle {\frac {\mathrm {d} \mathbf {A} ^{-1}}{\mathrm {d} t}}=-\mathbf {A} ^{-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{-1}.}
Similarly, if ϵ {\displaystyle \epsilon } is a small number then
( A + ϵ X ) − 1 = A − 1 − ϵ A − 1 X A − 1 + O ( ϵ 2 ) . {\displaystyle \left(\mathbf {A} +\epsilon \mathbf {X} \right)^{-1}=\mathbf {A} ^{-1}-\epsilon \mathbf {A} ^{-1}\mathbf {X} \mathbf {A} ^{-1}+{\mathcal {O}}(\epsilon ^{2})\,.}
More generally, if
d f ( A ) d t = ∑ i g i ( A ) d A d t h i ( A ) , {\displaystyle {\frac {\mathrm {d} f(\mathbf {A} )}{\mathrm {d} t}}=\sum _{i}g_{i}(\mathbf {A} ){\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}h_{i}(\mathbf {A} ),}
then,
f ( A + ϵ X ) = f ( A ) + ϵ ∑ i g i ( A ) X h i ( A ) + O ( ϵ 2 ) . {\displaystyle f(\mathbf {A} +\epsilon \mathbf {X} )=f(\mathbf {A} )+\epsilon \sum _{i}g_{i}(\mathbf {A} )\mathbf {X} h_{i}(\mathbf {A} )+{\mathcal {O}}(\epsilon ^{2}).}
Given a positive integer n {\displaystyle n} ,
d A n d t = ∑ i = 1 n A i − 1 d A d t A n − i , d A − n d t = − ∑ i = 1 n A − i d A d t A − ( n + 1 − i ) . {\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {A} ^{n}}{\mathrm {d} t}}&=\sum _{i=1}^{n}\mathbf {A} ^{i-1}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{n-i},\\{\frac {\mathrm {d} \mathbf {A} ^{-n}}{\mathrm {d} t}}&=-\sum _{i=1}^{n}\mathbf {A} ^{-i}{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\mathbf {A} ^{-(n+1-i)}.\end{aligned}}}
Therefore,
( A + ϵ X ) n = A n + ϵ ∑ i = 1 n A i − 1 X A n − i + O ( ϵ 2 ) , ( A + ϵ X ) − n = A − n − ϵ ∑ i = 1 n A − i X A − ( n + 1 − i ) + O ( ϵ 2 ) . {\displaystyle {\begin{aligned}(\mathbf {A} +\epsilon \mathbf {X} )^{n}&=\mathbf {A} ^{n}+\epsilon \sum _{i=1}^{n}\mathbf {A} ^{i-1}\mathbf {X} \mathbf {A} ^{n-i}+{\mathcal {O}}(\epsilon ^{2}),\\(\mathbf {A} +\epsilon \mathbf {X} )^{-n}&=\mathbf {A} ^{-n}-\epsilon \sum _{i=1}^{n}\mathbf {A} ^{-i}\mathbf {X} \mathbf {A} ^{-(n+1-i)}+{\mathcal {O}}(\epsilon ^{2}).\end{aligned}}}
Generalized inverse[edit]
Some of the properties of inverse matrices are shared by generalized inverses (e.g., the Moore–Penrose inverse), which can be defined for any m-by-n matrix.
Applications[edit]
For most practical applications, it is not necessary to invert a matrix to solve a system of linear equations; however, for a unique solution, it is necessary that the matrix involved be invertible.
Decomposition techniques like LU decomposition are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.
Regression/least squares[edit]
Although an explicit inverse is not necessary to estimate the vector of unknowns, it is unavoidable to estimate their precision, found in the diagonal of the posterior covariance matrix of the vector of unknowns.
Matrix inverses in real-time simulations[edit]
Matrix inversion plays a significant role in computer graphics, particularly in 3D graphics rendering and 3D simulations. Examples include screen-to-world ray casting, world-to-subspace-to-world object transformations, and physical simulations.
Matrix inverses in MIMO wireless communication[edit]
Matrix inversion also plays a significant role in the MIMO (Multiple-Input, Multiple-Output) technology in wireless communications. The MIMO system consists of N transmit and M receive antennas. Unique signals, occupying the same frequency band, are sent via N transmit antennas and are received via M receive antennas. The signal arriving at each receive antenna will be a linear combination of the N transmitted signals forming a NxM transmission matrix H. It is crucial for the matrix H to be invertible for the receiver to be able to figure out the transmitted information.
See also[edit]
Binomial inverse theorem
LU decomposition
Matrix decomposition
Matrix square root
Pseudoinverse
Singular value decomposition
Woodbury matrix identity
Notes[edit]
Jump up ^ Horn, Roger A.; Johnson, Charles R. (1985). Matrix Analysis. Cambridge University Press. p. 14. ISBN 978-0-521-38632-6. .
Jump up ^ Pan, Victor; Reif, John (1985), Efficient Parallel Solution of Linear Systems, Proceedings of the 17th Annual ACM Symposium on Theory of Computing, Providence: ACM
Jump up ^ Pan, Victor; Reif, John (1985), Harvard University Center for Research in Computing Technology Report TR-02-85, Cambridge, MA: Aiken Computation Laboratory
Jump up ^ "The Inversion of Large Matrices". Byte magazine. 11 (04): 181–190. April 1986.
Jump up ^ A proof can be found in the Appendix B of Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). "Superconducting quark matter in SU(2) color group". Zeitschrift für Physik A. 344: 99–115. doi:10.1007/BF01291027.
Jump up ^ Strang, Gilbert (2003). Introduction to linear algebra (3rd ed.). SIAM. p. 71. ISBN 0-9614088-9-8. , Chapter 2, page 71
Jump up ^ Bernstein, Dennis (2005). Matrix Mathematics. Princeton University Press. p. 44. ISBN 0-691-11802-7.
Jump up ^ Bernstein, Dennis (2005). Matrix Mathematics. Princeton University Press. p. 45. ISBN 0-691-11802-7.
Jump up ^ T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, Cambridge, MA, 2009, §28.2.
Jump up ^ Ran Raz. On the complexity of matrix product. In Proceedings of the thirty-fourth annual ACM symposium on Theory of computing. ACM Press, 2002. doi:10.1145/509907.509932.
Jump up ^ Stewart, Gilbert (1998). Matrix Algorithms: Basic decompositions. SIAM. p. 55. ISBN 0-89871-414-1.
Jump up ^ "A p-adic algorithm for computing the inverse of integer matrices". Journal of Computational and Applied Mathematics. 225: 320–322. doi:10.1016/j.cam.2008.07.044.
Jump up ^ "IML - Integer Matrix Library". cs.uwaterloo.ca. Retrieved 14 April 2018.
References[edit]
Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001) [1990]. "28.4: Inverting matrices". Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. pp. 755–760. ISBN 0-262-03293-7.
Further reading[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Inversion of a matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Matrix Mathematics: Theory, Facts, and Formulas at Google books
The Matrix Cookbook
External links[edit]
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Lecture on Inverse Matrices by Khan Academy
Linear Algebra Lecture on Inverse Matrices by MIT
LAPACK – a collection of FORTRAN subroutines for solving dense linear algebra problems
ALGLIB – includes a partial port of the LAPACK to C++, C#, Delphi, etc.
Online Inverse Matrix Calculator using AJAX
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text:
And death shall have no dominion
From Wikipedia, the free encyclopedia
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"And death shall have no dominion" is a poem written by Welsh poet Dylan Thomas (1914–1953). The title comes from St. Paul's epistle to the Romans (6:9).[1]
Contents
[hide]
1 Publication history
2 In popular culture
3 Notes
4 Bibliography
5 External links
Publication history[edit]
In early 1933 Thomas befriended Bert Trick, a grocer who worked in the Uplands area of Swansea.[2] Trick was an amateur poet who had several poems published in local papers. In spring 1933 Trick suggested the two men both write a poem on the subject of 'immortality'. Trick's poem, which was published in a newspaper the following year, contained the refrain "For death is not the end."[3] In 1933, in a notebook marked 'April', Thomas wrote the poem "And death shall have no dominion". Trick persuaded him to seek a publisher and in May of that year it was printed in New English Weekly.[3]
On 10 September 1936, two years after the release of his first volume of poetry (18 Poems), Twenty-five Poems was published. It revealed Thomas' personal beliefs pertaining to religion and the forces of nature, and included "And death shall have no dominion".
In popular culture[edit]
This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2017) (Learn how and when to remove this template message)
The poem is featured significantly in the television series Beauty and the Beast.
It was used at the start and ending of the movie Omega Doom.
The titles of the novels They Shall Have Stars by James Blish and No Dominion by Charlie Huston are taken from the poem.
In Brave Saint Saturn's album Anti-Meridian the final five lines of the first verse open the CD, accompanied by music.
In the film Truly, Madly, Deeply the title is quoted in a conversation about death.
In the film The Weight of Water from the book of the same title written by Anita Shreve, Sean Penn in the role of melancholy poet Thomas Janes recites the last four lines of the first stanza. At the end of the film after Janes drowns, the film reprises his recitation of the second and third lines of this section, but this time the film leaves the last line poignantly unspoken.
In the second and final part of the 2011 BBC TV miniseries The Field of Blood the poem's second through ninth lines are recited from memory by character Dr. Pete, played by Peter Capaldi, in a pub as he drunkenly faces his imminent death of cancer, seated alone.
George Clooney recites part of the poem in the 2002 movie Solaris.
In the 2003 film Rosenstrasse this poem is quoted as two Jewish women await deportation to Auschwitz.
In the Season 6 soundtrack of Lost the piece which shares its name with the poem is played while detailing Richard Alpert's life.
Notes[edit]
Jump up ^ Online copy of the poem
Jump up ^ Ferris (1989), p. 79
^ Jump up to: a b Ferris (1989) p. 83
Bibliography[edit]
Ferris, Paul (1989). Dylan Thomas, A Biography. New York: Paragon House. ISBN 1-55778-215-6.
External links[edit]
Bibliowiki has original media or text related to this article: And Death Shall Have No Dominion (in the public domain in Canada)
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"And death shall have no dominion"
"Do not go gentle into that good night"
"Fern Hill"
"In my Craft or Sullen Art"
"The Force That Through the Green Fuse Drives the Flower"
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Dominion (2016 film)
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Gather Ye Rosebuds While Ye May
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And death shall have no dominion
Der Erlkönig
Do not go gentle into that good night
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values: And death shall have no dominion – Czech

values: And death shall have no dominion – French

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