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)?