Q1. Let A be an n × n matrix and denote by ∥A∥F  the Frobenius norm of A. Recall that

the Frobenius norm has the following equivalent definition:

||A||F(2)  =工 |aij |2  = trace(AT A).

i,j

a) Show that ||A||F  = ||UA||F  for any orthogonal n × n matrix U .

b) Show that ||A||F  = ||AV ||F  for any orthogonal n × n matrix V .

Q2.  Conclude, with the help of Q1 above, that ||A||F  = 4 σi(2), where σ 1  ≥ σ2  ≥ ... ≥

σn  ≥ 0 are the singular values of A.

Q3. Let A ∈ Rn ×m , n ≥ m.

a) Use the SVD of A to deduce the SVD of AT A.

b) If m = n and A is full-rank, use a) to show that ∥AT A∥2  = ∥A∥2(2)  and κ2 (AT A) = κ2 (A)2 .

Q4. Work this exercise using pencil and paper. You can use MATLAB to check your work.

Let A be the following exterior (or outer) product of two vectors:

A =  「(l) −    · [ 1   1 ]

Note that A is 3 × 2. Answer the questions below; you can do so without ever forming A explicitly.

a) What is the rank r of A?

b) Think about the ”sum of rank one matrices” expression for the SVD of A, then consider the reduced SVD of A: A = Ur Σr VrT . What are the sizes of Ur , Σr , and Vr ?

c) Use the fact that the columns of Ur  and Vr  are orthonormal to figure out Ur ,Vr , and Σr .

Q5. Run the attached low rank approximation .m MATLAB code.