CSC338: Tutorial 5
1. Show that the two definitions of the induced matrix norm are equivalent, i.e.
max ||Ax|| = max ||Ax|| ||x||=1 ||x||≠0||x||
2. Show that for an n×n matrix A, its L1 norm is given by n
||A||1 =max|aij|
j
i=1
3. Suppose the matrix M is m×n where m > n, and M is full-rank. Show that A = MTM is symmetric positive definite.
4. Show that the Sherman-Morrison formula holds, that for an n × n invertible matrix A, and vectors u and v, we have
(A − uvT )−1 = A−1 + A−1uvT A−1 1−vTA−1u
Hint: multiply both sides by (A − uvT ).