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Matrix Computations

David Gleich

Purdue University

Fall 2017

Course number CS-51500

Tuesday and Thursday, 10:30-11:45am

Location Forney B124

Homework 2

Homework 2

Please answer the following questions in complete sentences in a clearly, typed prepared manuscript and submit the solution by the due date on Blackboard (Monday, September 11th, 2017, early in the morning)

Problem 0: Homework checklist

Problem 1: Prove or disprove

For the following questions, either prove that the statement is correct, or show a counter-example.

  1. The product of two diagonal matrices is diagonal.

  2. The product of two upper triangular matrices is upper triangular

  3. The product of two symmetric matrices is symmetric.

  4. The product of two orthogonal matrices is orthogonal.

  5. The product of two square, full rank matrices is full rank

Problem 2

There are a tremendous number of matrix norms that arise. An interesting class are called the orthgonally invariant norms. Norms in this class satisfy: for square orthogonal matrices and . Recall that a square matrix is orthogonal when , i.e. .

Show that is orthogonally invariant.

Problem 3

Consider the following function:

  1. Show that is a matrix norm. (Very easy!)

  2. Show that does not satisfy the sub-multiplicative property.

  3. Show that there exists such that: is a sub-multiplicative matrix-norm.

  4. Extra tough problem for the adventurous! Not graded. This problem has a relatively easy proof related to something we saw in class. But making it fully formal requires a few technicalities that are easy to get tripped up on. Now let be an arbitrary matrix norm. Show that there exists such that is a sub-multiplicative matrix-norm.

Problem 4

Let be a matrix norm and let be a real-valued vector. Consider the function:

  1. Show that is a vector norm.

  2. Show that if is a sub-multiplicative matrix norm, then the vector norm is consistent with the matrix norm. That is:

Problem 5 (Choice 1)

Note that you only have to do one of the two choices for problem 5.

In class we mentioned that the function is a vector norm. Use a computer to prepare a 3d plot of the unit-ball for this norm where . We may award up to 5 bonus points for an excellent graphic.

Problem 5 (Choice 2)

Note that you only have to do one of the two choices for problem 5.

Let be a vector norm. If (that is, is a scalar), show that , where is the absolute value function.