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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 257 39 "The Singular Value Decomp
osition (SVD):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 60 "For any rectangular matrix A, with real or complex entrie
s, " }}{PARA 0 "" 0 "" {TEXT -1 40 "there is a rectangular diagonal ma
trix  " }{XPPEDIT 18 0 "Sigma;" "6#%&SigmaG" }{TEXT -1 40 " containing
 only nonnegative real values" }}{PARA 0 "" 0 "" {TEXT -1 12 "(called \+
the " }{TEXT 256 15 "singular values" }{TEXT -1 55 " of A), and square
 unitary matrices U and V\n(for which " }{XPPEDIT 18 0 "V^H;" "6#)%\"V
G%\"HG" }{TEXT -1 8 " is the " }{TEXT 258 20 "Hermitian transpose," }
{TEXT -1 37 " i.e., the conjugated transpose of V)" }}{PARA 0 "" 0 "" 
{TEXT -1 43 "so that A can be decomposed as the product:" }}{PARA 0 "
" 0 "" {TEXT -1 9 "         " }{XPPEDIT 18 0 "A = U*Sigma*V^H;" "6#/%
\"AG*(%\"UG\"\"\"%&SigmaGF')%\"VG%\"HGF'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "?Svd" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "?ht
ranspose" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n := 3;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"nG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "A := randmatrix(n,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-
%'matrixG6#7%7%\"#x\"#m\"#a7%!\"&\"#**!#h7%!#]!#7!#=" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "rank(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Singular_values \+
:= evalf(Svd(A));\n# Maple insists this has to be done with \"evalf\" \+
outside" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0Singular_valuesG-%'vecto
rG6#7%$\"+/#=$>8!\"($\"+*zBL4\"F+$\"+4M#z)=!\")" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 23 "The singular values of " }{XPPEDIT 18 0 "A;" "6#%\"A
G" }{TEXT -1 37 " are essentially the eigenvalues of  " }{XPPEDIT 18 
0 "A*A^T;" "6#*&%\"AG\"\"\")F$%\"TGF%" }{TEXT -1 1 " " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "map( x->sqrt(Re(evalf(x))),\n    al
lvalues([eigenvals( evalm(  A &* transpose(A)  ) )])\n);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7%$\"+/#=$>8!\"($\"+/M#z)=!\")$\"+*zBL4\"F&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "We can also get the U and V matric
es of the SVD as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "Singular_values := evalf(Svd(A,U,V));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%0Singular_valuesG-%'vectorG6#7%$\"+/#=$>8!\"($\"+*zBL
4\"F+$\"+4M#z)=!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Turn the s
equence of singular values into a diagonal matrix." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 51 "Sigma := diag( seq(Singular_values[i_],i_=1
..n) );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&SigmaG-%'matrixG6#7%7%
$\"+/#=$>8!\"(\"\"!F-7%F-$\"+*zBL4\"F,F-7%F-F-$\"+4M#z)=!\")" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalm( A - U &* Sigma &* htr
anspose(V) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%$\"\"
\"!\")$\"\"$F*$!\"#F*7%$\"\"'!\"*F-$\"\"#F*7%\"\"!$!\"\"F*F6" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "U is unitary." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 9 "evalm(U);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'matrixG6#7%7%$!+t9(*Rx!#5$\"+#zNFw%F*$\"+p?VsTF*7%$!+[5//cF*$!+;P0?
#)F*$!+E0k75F*7%$\"+XwYZHF*$!+P*G?7$F*$\"+:[NJ!*F*" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 7 "det(U);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+++++5!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm(U &* \+
htranspose(U));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%$\"
+**********!#5$!\"&!#6$\"\"\"F*7%F+$\"+++++5!\"*$\"#5F-7%F.F4F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "V is unitary also." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(V);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%$!+5Y(>U&!#5$\"+7I'z:&F*$!+T$pIj'F*7%$!
+l&y_M)F*$!+9uYDUF*$\"++:yNNF*7%$!+g\")Q!z*!#6$\"+\"*Gd_uF*$\"+?:]&f'F
*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(V);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+++++5!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "evalm(V &* htranspose(V));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%$\"+)*********!#5\"\"!F+7%F+$\"+++++5!
\"*F+7%F+F+F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "We can validate \+
that the Singular Value Decomposition really holds for A:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalm(A - U &* Sigma &* htranspose(
V));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%$\"\"\"!\")$\"
\"$F*$!\"#F*7%$\"\"'!\"*F-$\"\"#F*7%\"\"!$!\"\"F*F6" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 41 "evalm( Sigma - htranspose(U) &* A &* V );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!$!#<!\"*$\"#=
F+7%$!\"#F+F(F(7%$\"\"#F+$\"\"*F+F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 }