lecture6 - EECS 22} A LECTURE (3 Cadiz/1 5' babel/L...

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Unformatted text preview: EECS 22} A LECTURE (3 Cadiz/1 5' babel/L APFQHOUX A (Q Self — Adl‘oini mags C’Hvet/L (H)TF,<-,->H)) a H[letSp’JOL(_L. Let A44 —> H be, 6L OWN/invous (1mm my) VON/14 adjoth A" .~ H —>H. We .3an Wed \H/UL maf/L is Selj-adfomt A=A*) or Qauivatm‘tbj) 4X>A7>H= (AXJDH VX’VQH- QXQMEUL (HQ/Imhhom mmhn'wD LEJC H=J¥n and Let A La refranfid bj 0L Wuflrvix A: (acj')2‘d'6§|..n§ 6 FM“. Wm A is gelf—aoljo'mt mdflx A jg Henmbhou/L > or Qfioivaflm ) A=A*) \‘ W1me an- = E; Vid 9 {fun} 5 Orv/140515 A is eqv +0 H1 Can/Lynx cmflgafi Lawoaé. fidfl [Jim/{+an maHIX A mahm U 6 IFm‘n )S SOL/La 7L0 [76, Uni?!"an U*U : UU* = In (equivath )\H/\L yL Calumm € n VOVU§ of U mm or anmmafl basw of AM). If AF: ,wah ammhm )5 COLULi OYWog/emafl. G. 1 The Singular-Value Decomposition Theorem Let All 6 me" with rank (M) = r. Then we can find unitary matrices U E mem and V 6 CM" such that _ xx _ El 0 =k M — UEV — U [ 0 0 J V where 0'1 0 0 0 0'2 0 $1: . . . O O or The real numbers 01 Z 02 Z Z or > O are called the singular values ofM and the representation above is called the singular-value decomposition of M. Theorem Let M E me" with rank (A!) = r and let M 2 UEV" be the singular— value decomposition ofM. Partition U and V as U=[U1U2],V=[V1 v2] where U1 and V1 are in CT”. Then (a) The columns of U1 and U2 form orthonormal bases for "RM/1’) ana’ N[M’") re- spectiuely. (b) The columns of V1 and V2 form orthonormal bases for R(M*) ana’ re— spectiuely. Computing the singular-value decomposition While definitely not the method of choice uis-a—uis numerical aspects, the following result provides an adequate method for determining the singular—value decomposition of a matrix. Theorem Let 1M 6 me" with rank (M) = r. Let A1,---,/\r be the nonzero eigenvalues of M*l\/L These will be nonnegatiue because JVPM Z 04 Also, from item {F-4) it follows that there exist unitary matrices U E mem and V 6 CM" such that A 0 Vi? o 0 l A 0 00]U ana’ MM=V[ M’M" = U[ where A = diag (A1, - - - ,Ar). —l>OOll0k Then the singular—value decomposition ofM 2's 1 M=U[A2 0]v* 0 0 l i and the singular values of I” are A2 , - - - , A3. 4 Optimal rank q approximations of matrices Theorem Let [M E CmX" with rank (M) = T have singulat value decomposition as 01 0 0 0 a 0 A/IzU 21 0 V“, where 21: ,2 , 0 0 ; 0 0 or Define the matrix 2 0 01 0 0 A __ I It A __ - M—U[0 0]V,where 21— : 0 0 oq ...
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This note was uploaded on 04/11/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.

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lecture6 - EECS 22} A LECTURE (3 Cadiz/1 5' babel/L...

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