Problem 1: Prove that any real-valued matrix has a real-valued SVD, i.e. an SVD factorization
where
and
are real-valued matrices. Hint: Do not re-do the existence uniqueness proof we did in class with a restriction to real matrices. Start with a possibly complex SVD
(where
and
are possibly complex but
is necessarily real), and use this to show there must be real-valued unitary
and
that also form an SVD
. I suspect it will be helpful to express the SVDs as sums over columns of
and
, i.e.
, as shown in problem 3.)
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 = 0.8 0.2
(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.
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
. You can do this measuring in the units of your ruler without worrying about coordinating those units with the unit length on your plot.
(f) Now compute numerically and plot it on the
plot. Does it match what you came up with in (e)?