Problem

1. Suppose that A is an invertible matrix with eigenvector v. Show that v is also an eigenvector for A-1.

2. Show that the eigenvalues of MMT are the same as that of MTM. Are their eigenvectors also the same?

3. Compare the speed of a library function for matrix multiplication to your own implementation of the nested loops algorithm.

• How much faster is the library on products of random n × n matricies, as a function of n as n gets large?

• What about the product of an n × m and m × n matrix, where n « m?

• By how much do you improve the performance of your implementation to calculate C = A · B by first transposing B internally, so all dot products are computed along rows of the matrices to improve cache performance?