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) 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.