Hi.

problem:For each of the following, verify the inequality and give an example of a nonzero vector or matrix for which equality is achieved. In this problem x is an m-vector and A is an matrix.

(a) ,

(b) ,

(c) ,

(d) .

attempt:(a) I rewrite the inequality so it is easier for me to read:

Let be the largest component of . Then .

If the remaining components are zero, the inequality becomes an equality. If one or more of the remaining components are nonzero, it becomes an inequality.

Example of vector: .

I'm sure there are better ways to verify this.

(b) Again, I rewrite:

.

Let be the largest component of .

Since (m-times) and since for are smaller than , the inequality holds.

It's an equality for all vectors with

I am not at all sure what to do with (c) and (d).

Any hints are appreciated.