|
|
Re: [MATHEDU] SVD and other acronyms
Posted:
Feb 26, 2001 11:58 AM
|
|
On Mon, 26 Feb 2001, David Epstein wrote:
> Matthias Kawski wrote:
> >This student has every right to "accept" the working of SVD, ...
>
> Am I the only one on this list who is totally mystified by the acronym
> SVD? Or is it just like a class of students where those who understand
> don't need to ask, and those who don't understand are too embarrassed
> to ask? .....
Dear David:
Thank you for this reminder -- all too often am I culpable of such
oversights. But in this case the original mention of SVD was in:
Date: Wed, 21 Feb 2001 10:56:40 -0600
From: Jerry Uhl <juhl@cm.math.uiuc.edu>
To: mathedu@warwick.ac.uk
Subject: [MATHEDU] Engineer's view of proofs
Nonetheless, here is a brief synopsis of SVD as I see it,
followed by a short opinion-piece that explains why SVD
(or familiarity with SVD, or inclusion of SVD) serves well
to differentiate between different linear algebras and
different mathematicians"....
SVD stands for singular-value-decomposition (of a matrix or
linear operator). In spirit it is similar to diagonalization
of a matric A which tries to find an invertible matrix U
such that U * A * Uinverse is diagonal.
SVD aimes at finding orthogonal matrices U and V together
with a diagonal matrix S with nonnegative entries such that
A = U * S * V. Note that this makes sense also in the case
where A in not square (i.e. for operators whose domain and
range do not agree). A very common use is approximation/
compression: Form a new matrix S2 by replacing all entries
of S that are smaller than some threshold to zero. Finally,
instead of working with the full matrix A, only "remember",
save or "transmit" these nonzero entries of S2 together with
the associated orthogonal rows/columns of U and V. See
http://math.la.asu.edu/~kawski/MATLAB/matlab.html#compression
for sample images and associated implementations into MATLAB
(ready to be used in class). To calculate the SVD of a matrix
A, start with the product of A with its transpose. Since this
is symmetric it can be orthogonally diagonalized using com-
paratively efficient numerical algorithms (very different from
calculating characteristic polynomials via determinants etc.)
There is significant tension between many of our linear algebra
courses in math departments and applied users. The contrast of
SVD versus Jordan normal form (which is often considered "the
best" one can do for matrices that cannot be diagonalized due
to repeated roots with rank deficient eigenspaces) epitomizes
this tension: The linear algebra course that I took as a student,
and which I taught in my earlier years aimed straight at the
Jordan canonical form as the final goal.
Later I got Gil Strang's LineAlgebra text into my hands, which
was a real eye-opener for me. Its clear goal is the SVD,
constantly prepared for by the development of "the four
fundamental subspaces" (range and kernel of A and of its
transpose).
When forced to make a decision whether to use MATLAB (primarily a
numerical package) or a computer algebra system for the next
linear algebra class that I was to teach, it again came down to:
"Are we going for SVD or for Jordan as the class' final goal?"
(It was special section that was part of an integrated engineering
sophomore course, but it still was the math deprtment's course...)
Indeed, a full text seach of all MATLAB files on my disk revealed
no occurrence at all of the string "jordan". Initially baffled by
this, I soon recognized that this made sense as two (even finite-
precision) "floating point numbers" have a negligible chance of
being the same... Numerically, repeated roots are a nonissue!
On the other hand, many textbooks work with "small matrices with
small integer entries" -- in that case a computer algebra system
is the right choice! [[A beautiful experience of the properties
of such small-integer-matrices is provided by the MATLAB file
catt.m, see
http://math.la.asu.edu/~kawski/MATLAB/matlab.html#lintrafo
which "randomly" creates small integer 2x2 matrices and
overlays the image of a "cat's face" under the associated
linear transformation with the original face. [The original
purpose was to encapsulate eigenvalues/vectors in a visual
experience that could serve also as the "logo" for my class].
Now, repeated roots and zero eigenvalues have a large nonzero
probability, -- "surprisingly often" the image-face is a
line segment!
Matthias
**********************************************************
Matthias Kawski http://math.la.asu.edu/~kawski
Department of Mathematics e-mail: kawski@asu.edu
Arizona State University office: (480) 965 3376
Tempe, Arizona 85287-1804 FAX: (480) 965 0461
NOTE: NEW AREA CODES FOR PHOENIX home: (480) 893 0107
**********************************************************
This is an unmoderated distribution list discussing teaching and learning
of post-calculus mathematics.---David.Epstein@warwick.ac.uk
Get guidelines before posting: email majordomo@warwick.ac.uk saying
get mathedu guidelines
(Un)subscribe to mathedu(-digest)by email to majordomo@warwick.ac.uk saying:
(un)subscribe mathedu(-digest)
|
|
