# Diagonalisability problem and others

rainwyz0706

## Homework Statement

1.Prove that if A is a real matrix then At A is diagonalisable.

2. Given a known 3*3 matrix A, Calculate the maximum and minimum values of ||Ax|| on the sphere ||x|| = 1.

## The Attempt at a Solution

For the first problem, I'm thinking of proving that AtA is symmetric, but I'm not sure which properties to use.
For the second one, is X a 3*n matrix? Do I need to discuss n?
Any help is greatly appreciated!

Staff Emeritus
Homework Helper
For the first problem, I'm thinking of proving that AtA is symmetric, but I'm not sure which properties to use.
If $B=A^TA$, write down what $B_{ij}$ and $B_{ji}$ are in terms of the elements of A and show that they're equal.

rainwyz0706
Thanks a lot! I can't believe I missed it in the first place.
Could anyone give me some hints about problem 2?

Homework Helper
For the second one, yes, x must be such a vector that the product Ax is meaningful, i.e. a 3x1 vector (matrix).

rainwyz0706
thx, I finished it!