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gibson:teaching:fall-2016:iam961:hw2

# IAM 961 HW2, fall 2016

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)?