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Re: [MATHEDU] SVD and other acronyms
Posted:
Feb 27, 2001 1:05 PM
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Introduction to SVD in 2 dimensions:
Take a 2 by 2 matrix A.
When you multiply every point of the unit circle centered at {0,0},
you get an ellipse.
The SVD stretch factors for A are the lengths stretch1 and stretch2
of the semi axes of that ellipse.
Then you get two unit vectors a1 and a2 so that A.a1 is one of the
semiaxes and A.a2 is the other.
With a little thought, you can see that
A =
Matrix with A.a1/stretch1 and A.a2/stretch2 in the columns)
(matrix with stretch1 and stretch2 on diagonal) (Matrix
with rows a1 and a2).
That's the SVD.
The reason this works well for students is that each of the three
matrices has a distinctive geometric action
-Jerry
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Jerry Uhl juhl@cm.math.uiuc.edu
Professor of Mathematics, University of Illinois at Urbana-Champaign
Member, Mathematical Sciences Education Board of National Research Council
Calculus&Mathematica, Vector Calculus&Mathematica,
DiffEq&Mathematica, Matrices,Geometry&Mathematica, NetMath
http://www-cm.math.uiuc.edu , http://netmath.math.uiuc.edu, and
http://matheverywhere.com
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