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Subject: SVD
Date: Wed, 28 Nov 2007 09:57:00 +0800
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<HTML><HEAD><TITLE>SVD</TITLE>
<META http-equiv=3DContent-Type content=3D"text/html; =
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<META content=3D"Introduction to Singular Value Decomposition" =
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<META content=3D"singular value decomposition, svd, introduction to svd" =
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<BODY vLink=3D#666666 aLink=3D#999999 link=3D#000066 bgColor=3D#ffffff =
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topMargin=3D8> =20
<TABLE cellSpacing=3D0 cellPadding=3D5 border=3D1>
<TBODY>
<TR>
<TD vAlign=3Dtop width=3D95 bgColor=3D#99ff99 rowSpan=3D2>
<HR width=3D"100%">
<A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/index.html">Introduction</A=
> =20
<HR width=3D"100%">
<A =
href=3D"http://www.uwlax.edu/faculty/will/svd/action/index.html">Matrix=20
Action</A> =20
<HR width=3D"100%">
<A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/perpframes/index.html">Perp=
frames,=20
Aligners and Hangers</A> =20
<HR width=3D"100%">
<A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/stretchers/index.html">Stre=
tchers</A> =20
<HR width=3D"100%">
<A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/coordinates/index.html">Coo=
rdinates</A> =20
<HR width=3D"100%">
<A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/projections/index.html">Pro=
jections</A> =20
<HR width=3D"100%">
<A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/svd/index.html">SVD</A>&nbs=
p;=20
<HR width=3D"100%">
<A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/subspaces/index.html">Matri=
x=20
Subspaces </A> =20
<HR width=3D"100%">
<A =
href=3D"http://www.uwlax.edu/faculty/will/svd/systems/index.html">Linear =
Systems, Pseudo-Inverse</A> =20
<HR width=3D"100%">
<A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/condition/index.html">Condi=
tion=20
Number</A> =20
<HR width=3D"100%">
<A =
href=3D"http://www.uwlax.edu/faculty/will/svd/norm/index.html">Matrix=20
Norm, Rank One</A> =20
<HR width=3D"100%">
<A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/compression/index.html">Dat=
a=20
Compression</A> =20
<HR width=3D"100%">
<A =
href=3D"http://www.uwlax.edu/faculty/will/svd/noise/index.html">Noise=20
Filtering</A> =20
<HR width=3D"100%">
<H6>Todd Will<BR>UW-La Crosse</H6></TD>
<TD vAlign=3Dcenter align=3Dmiddle width=3D10 =
bgColor=3D#cccccc> <A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/svd/index.html#Two-thirds">=
Two-thirds=20
Theorem =20
<HR width=3D"100%">
</A><A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/svd/index.html#Theorem: If =
A is an m x n matrix, then there is">Good=20
Basis Theorem</A></TD>
<TD vAlign=3Dtop align=3Dmiddle bgColor=3D#ccffff>
<CENTER>
<H1>SVD</H1></CENTER></TD>
<TD vAlign=3Dcenter align=3Dmiddle width=3D10 bgColor=3D#cccccc><A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/svd/index.html#Theorem =
SVD">SVD =20
<HR width=3D"100%">
</A><A=20
=
href=3D"http://www.uwlax.edu/faculty/will/svd/svd/index.html#Exercises">E=
xercises</A></TD></TR>
<TR>
<TD bgColor=3D#ffffff colSpan=3D3>
<H2><FONT color=3D#ff0000>Singular Value =
Decomposition</FONT></H2>The=20
singular value decomposition for a matrix A writes A as a product=20
(hanger)(stretcher)(aligner). <BR> =20
<P>It's an amazing and useful fact that every m x n matrix has a =
singular=20
value decomposition. =20
<P>The following theorem goes two-thirds of the way to proving =
this=20
fact: =20
<HR width=3D"100%">
<H2><A name=3DTwo-thirds></A><FONT color=3D#ff0000>Two-thirds=20
Theorem</FONT></H2>For an <IMG height=3D17 =
alt=3D[Graphics:svdgr1.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr1.gif" =
width=3D31=20
align=3DABSCENTER> matrix <IMG height=3D17 =
alt=3D[Graphics:svdgr2.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr2.gif" =
width=3D72=20
align=3DABSCENTER> and any orthonormal basis <IMG height=3D17 =
alt=3D[Graphics:svdgr3.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr3.gif" =
width=3D91=20
align=3DABSCENTER> of <IMG height=3D17 =
alt=3D[Graphics:svdgr4.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr4.gif" =
width=3D19=20
align=3DABSCENTER>, =20
<P>define <IMG height=3D17 alt=3D[Graphics:svdgr5.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr5.gif" =
width=3D61=20
align=3DABSCENTER> =20
<P>and =20
<P><IMG height=3D49 alt=3D[Graphics:svdgr6.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr6.gif" =
width=3D128=20
align=3DABSCENTER>. =20
<P>Then <IMG height=3D77 alt=3D[Graphics:svdgr7.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr7.gif" =
width=3D240=20
align=3DABSCENTER>. =20
<P><B><FONT color=3D#3333ff>Proof</FONT>:</B> Using first the row =
way and=20
then the column way to multiply a matrix times a point, you see =
that the=20
right hand side of the equation sends <IMG height=3D17=20
alt=3D[Graphics:svdgr8.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr8.gif" =
width=3D15=20
align=3DABSCENTER> to <IMG height=3D35 =
alt=3D[Graphics:svdgr9.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr9.gif" =
width=3D132=20
align=3DABSCENTER>. =20
<P>Thus the two sides of the equation agree on the basis <IMG =
height=3D17 alt=3D[Graphics:svdgr10.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr10.gif" =
width=3D89=20
align=3DABSCENTER> and so must be equal. =20
<P>
<HR width=3D"100%">
<P>The two-thirds theorem gets you two-thirds of the way to the =
SVD. =20
<P>It says that given any orthonormal basis <IMG height=3D17=20
alt=3D[Graphics:svdgr11.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr11.gif" =
width=3D91=20
align=3DABSCENTER> of <IMG height=3D17 =
alt=3D[Graphics:svdgr12.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr12.gif" =
width=3D19=20
align=3DABSCENTER> you can write =20
<CENTER><IMG height=3D77 alt=3D[Graphics:svdgr13.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr13.gif" =
width=3D240=20
align=3DABSCENTER> </CENTER>
<CENTER><IMG height=3D19 alt=3D[Graphics:svdgr14.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr14.gif" =
width=3D207=20
align=3DABSCENTER>. </CENTER>
<CENTER> </CENTER>
<P>So you've got the stretcher and the aligner -- if <IMG =
height=3D19=20
alt=3D[Graphics:svdgr15.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr15.gif" =
width=3D100=20
align=3DABSCENTER> were a hanger matrix then this would be a =
Singular Value=20
Decomposition for <I>A</I>. =20
<P>For <IMG height=3D19 alt=3D[Graphics:svdgr16.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr16.gif" =
width=3D100=20
align=3DABSCENTER> to be a hanger matrix requires that the =
columns <IMG=20
height=3D35 alt=3D[Graphics:svdgr17.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr17.gif" =
width=3D68=20
align=3DABSCENTER> be pairwise perpendicular. =20
<P>So one challenge to finding an SVD for <I>A</I> is to find an=20
orthonormal basis of <IMG height=3D17 =
alt=3D[Graphics:svdgr18.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr18.gif" =
width=3D19=20
align=3DABSCENTER>, <IMG height=3D17 =
alt=3D[Graphics:svdgr19.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr19.gif" =
width=3D61=20
align=3DABSCENTER> so that for all <IMG height=3D17=20
alt=3D[Graphics:svdgr20.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr20.gif" =
width=3D28=20
align=3DABSCENTER>, <IMG height=3D19 =
alt=3D[Graphics:svdgr21.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr21.gif" =
width=3D73=20
align=3DABSCENTER>. =20
<P>
<HR width=3D"100%">
<H3><A name=3D"Theorem: If A is an m x n matrix, then there =
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