Problem 1: Prove that any real-valued matrix has a real-valued SVD, i.e. an SVD factorization
where
and
are real-valued matrices.
Problem 2: Show that if is m x n and
is p x n, the product
can be written as
where and
are the columns of
and
. This is more a matter of understanding and proper use
of notation than any kind of deep proof. It should take about three lines.
Problem 3: Let be an m x n matrix with SVD
. By applying the results of problem 2 to the matrices
and
, show that
where . For simplicity you can assume
so that
. Note that if
is the number of nonzero singular values (or equivalently, the rank of
), then clearly one can also write the sum going to just
instead of
.
Problem 4: Continuing from problem 3, let
where . Show that
is the closest rank-
approximation to
in the 2-norm, i.e. that
(where if
). This is Theorem 5.8 in Trefethen & Bau. You can follow that proof, just write it out in your own words, improving on presentation & argument where you can.
Problem 5: Given the following 2 x 2 matrix and 2-vector
,
A = -0.0954915028125262 -1.2449491424413903 0.6571638901489170 0.7135254915624212 x = 1.50000000000000 1.00000000000000
(a) compute the SVD .
(b) make two plots, one showing the columns of of
as an orthogonal basis for the domain of
, and another showing the columns of
of
as the same for the range of
.
(c) superimpose a unit circle on the
plot.
(d) superimpose the image of that unit circle under on the
plot. I.e. for
plot
. What can you say about the image of
under
in relation to the SVD?
(e) Plot the vector on the
plot
x = 1.50000000000000 1.00000000000000
Now figure out where will be on the
plot geometrically using the SVD, as follows
Following this formula, measure (with a ruler!) the components of along
and
. Multiply those lengths by the scaling factors
and
. Then measure out the scaled lengths along
and
, and add those components together to get the position of the vector
. Do this all by eye or with a ruler.
(f) Now compute numerically and plot it on the
plot. Does it match what you came up with in (e)?