📄 svd.mht
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width=3D174=20
align=3DABSCENTER> and see that <BR> =20
<UL>
<LI>no matter what <IMG height=3D17 =
alt=3D[Graphics:svdgr76.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr76.gif" =
width=3D11=20
align=3DABSCENTER> you choose <IMG height=3D17=20
alt=3D[Graphics:svdgr77.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr77.gif" =
width=3D32=20
align=3DABSCENTER> is a unit vector in <IMG height=3D17=20
alt=3D[Graphics:svdgr78.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr78.gif" =
width=3D17=20
align=3DABSCENTER> =20
<LI><IMG height=3D18 alt=3D[Graphics:svdgr79.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr79.gif" =
width=3D201=20
align=3DABSCENTER>. </LI></UL> =20
<P>Since <IMG height=3D17 alt=3D[Graphics:svdgr80.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr80.gif" =
width=3D17=20
align=3DABSCENTER> is a unit vector in <IMG height=3D17=20
alt=3D[Graphics:svdgr81.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr81.gif" =
width=3D17=20
align=3DABSCENTER> maximizing <IMG height=3D17 =
alt=3D[Graphics:svdgr82.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr82.gif" =
width=3D49=20
align=3DABSCENTER> and <IMG height=3D17 =
alt=3D[Graphics:svdgr83.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr83.gif" =
width=3D32=20
align=3DABSCENTER> is in <IMG height=3D17 =
alt=3D[Graphics:svdgr84.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr84.gif" =
width=3D17=20
align=3DABSCENTER> for all <IMG height=3D17 =
alt=3D[Graphics:svdgr85.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr85.gif" =
width=3D11=20
align=3DABSCENTER>, you know that <IMG height=3D17=20
alt=3D[Graphics:svdgr86.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr86.gif" =
width=3D112=20
align=3DABSCENTER> has a maximum at <IMG height=3D17=20
alt=3D[Graphics:svdgr87.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr87.gif" =
width=3D32=20
align=3DABSCENTER>. <BR> =20
<P>This tells you <IMG height=3D17 =
alt=3D[Graphics:svdgr88.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr88.gif" =
width=3D60=20
align=3DABSCENTER>. <BR> =20
<P>Now compute: <BR><IMG height=3D17 =
alt=3D[Graphics:svdgr89.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr89.gif" =
width=3D112=20
align=3DABSCENTER> <BR> =20
<P><IMG height=3D17 alt=3D[Graphics:svdgr90.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr90.gif" =
width=3D106=20
align=3DABSCENTER> <BR> =20
<P><IMG height=3D18 alt=3D[Graphics:svdgr91.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr91.gif" =
width=3D333=20
align=3DABSCENTER> <BR> =20
<P><IMG height=3D38 alt=3D[Graphics:svdgr92.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr92.gif" =
width=3D341=20
align=3DABSCENTER> <BR> <BR> =20
<P>When you remember <IMG height=3D17 =
alt=3D[Graphics:svdgr93.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr93.gif" =
width=3D75=20
align=3DABSCENTER>, <IMG height=3D18 =
alt=3D[Graphics:svdgr94.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr94.gif" =
width=3D75=20
align=3DABSCENTER>, and <IMG height=3D18 =
alt=3D[Graphics:svdgr95.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr95.gif" =
width=3D75=20
align=3DABSCENTER> are just numbers, you understand that its =
nothing more=20
than tedious to compute: =20
<P><IMG height=3D38 alt=3D[Graphics:svdgr96.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr96.gif" =
width=3D417=20
align=3DABSCENTER> <BR> =20
<P>Plugging in <IMG height=3D17 alt=3D[Graphics:svdgr97.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr97.gif" =
width=3D32=20
align=3DABSCENTER> gives you <IMG height=3D18 =
alt=3D[Graphics:svdgr98.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr98.gif" =
width=3D126=20
align=3DABSCENTER>. <BR> =20
<P>But you already know that <IMG height=3D17=20
alt=3D[Graphics:svdgr99.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr99.gif" =
width=3D60=20
align=3DABSCENTER>, so after canceling the <IMG height=3D17=20
alt=3D[Graphics:svdgr100.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr100.gif" =
width=3D11=20
align=3DABSCENTER> you get <IMG height=3D18 =
alt=3D[Graphics:svdgr101.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr101.gif" =
width=3D85=20
align=3DABSCENTER> which is just what you wanted. =20
<P>
<HR width=3D"100%">
<H3><FONT color=3D#3333ff>Proof 3: Based on the spectral=20
theorem</FONT></H3>This proof is slick IF YOU'VE ALREADY SEEN THE =
SPECTRAL=20
THEOREM. =20
<P>If you haven't seen the spectral theorem, then skip this =
proof. =20
<P>Given <IMG height=3D17 alt=3D[Graphics:svdgr102.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr102.gif" =
width=3D72=20
align=3DABSCENTER> and an orthonormal basis <IMG height=3D17=20
alt=3D[Graphics:svdgr103.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr103.gif" =
width=3D89=20
align=3DABSCENTER> of <IMG height=3D17 =
alt=3D[Graphics:svdgr104.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr104.gif" =
width=3D19=20
align=3DABSCENTER> , =20
<P><IMG height=3D19 alt=3D[Graphics:svdgr105.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr105.gif" =
width=3D125=20
align=3DABSCENTER> =20
<P>iff <IMG height=3D19 alt=3D[Graphics:svdgr106.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr106.gif" =
width=3D136=20
align=3DABSCENTER> =20
<P>iff <IMG height=3D19 alt=3D[Graphics:svdgr107.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr107.gif" =
width=3D78=20
align=3DABSCENTER> =20
<P>iff <IMG height=3D17 alt=3D[Graphics:svdgr108.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr108.gif" =
width=3D89=20
align=3DABSCENTER> are all eigenvectors of <IMG height=3D17=20
alt=3D[Graphics:svdgr109.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr109.gif" =
width=3D26=20
align=3DABSCENTER>. =20
<P><B><FONT color=3D#3333ff>Conclusion:</FONT></B> The desired =
basis is=20
guaranteed by spectral theorem since <IMG height=3D17=20
alt=3D[Graphics:svdgr109.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr109.gif" =
width=3D26=20
align=3DABSCENTER> is symmetric. =20
<P>
<HR width=3D"100%">
<H3><A name=3D"Theorem SVD"></A><FONT color=3D#ff0000>Theorem: =
Every matrix=20
has a singular value decomposition. </FONT></H3>The theorem =
above=20
almost gives you the SVD for any matrix. =20
<P>The only problem is that although the columns of the "hanger" =
matrix=20
are pairwise perpendicular, they might not form a basis =
for <IMG=20
height=3D17 alt=3D[Graphics:svdgr110.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr110.gif" =
width=3D21=20
align=3DABSCENTER>. =20
<P>For example, suppose for a 5x4 matrix <IMG height=3D18=20
alt=3D[Graphics:svdgr111.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr111.gif" =
width=3D66=20
align=3DABSCENTER> the procedure outlined above gives you: =20
<CENTER><IMG height=3D75 alt=3D[Graphics:svdgr112.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr112.gif" =
width=3D237=20
align=3DABSCENTER>. </CENTER> =20
<P>To complete the decomposition, let <IMG height=3D20=20
alt=3D[Graphics:svdgr113.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr113.gif" =
width=3D78=20
align=3DABSCENTER> be an orthonormal basis for the three =
dimensional=20
subspace of <IMG height=3D18 alt=3D[Graphics:svdgr114.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr114.gif" =
width=3D19=20
align=3DABSCENTER> perpendicular to <IMG height=3D20=20
alt=3D[Graphics:svdgr115.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr115.gif" =
width=3D69=20
align=3DABSCENTER>. <BR> =20
<P>Then write =20
<CENTER><IMG height=3D75 alt=3D[Graphics:svdgr116.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr116.gif" =
width=3D237=20
align=3DABSCENTER> </CENTER>
<CENTER><IMG height=3D76 alt=3D[Graphics:svdgr117.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr117.gif" =
width=3D271=20
align=3DABSCENTER> </CENTER>
<CENTER> </CENTER>(1) The two sides agree on the =
basis <IMG=20
height=3D17 alt=3D[Graphics:svdgr118.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr118.gif" =
width=3D102=20
align=3DABSCENTER>. <BR> <BR> =20
<P>This, finally, is a singular value decomposition for=20
<I>A. </I> =20
<P>
<HR width=3D"100%">
<BR> =20
<P><FONT color=3D#3333ff><B>Comments</B>: </FONT> =
<BR> =20
<UL>
<LI>The diagonal entries of the stretcher matrix are called the=20
"singular values of <I>A</I>". <BR> =20
<LI>An extra row of zeros has been added to the stretcher matrix =
to=20
produce the dimensions required for the multiplication. If =
<I>A</I> is m=20
x n with <IMG height=3D17 alt=3D[Graphics:svdgr119.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr119.gif" =
width=3D34=20
align=3DABSCENTER>, then rows will be deleted. =20
<P>In either case, the dimensions of the stretcher matrix will =
always=20
match the dimensions of <I>A</I>. <BR> </P>
<LI>The decomposition shows that the action of every matrix can =
be=20
described as a rotation followed by a stretch followed by =
another=20
rotation. <BR> =20
<LI>The proofs above are meant to show that every matrix has an=20
SVD. You can compute SVD's for mx2 matrices by hand, but =
you=20
should use a machine to compute SVD's for bigger matrices.=20
</LI></UL> =20
<P>
<HR width=3D"100%">
<H2><A name=3DExercises></A><FONT =
color=3D#ff0000>Exercises</FONT></H2>1.=20
Above, you saw that if A is a <IMG height=3D17=20
alt=3D[Graphics:svdgr120.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr120.gif" =
width=3D23=20
align=3DABSCENTER> matrix <IMG height=3D18 =
alt=3D[Graphics:svdgr121.gif]=20
src=3D"http://www.uwlax.edu/faculty/will/svd/svd/svdgr121.gif" =
width=3D66=20
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