# Diagonalisability problem and others

1. Feb 22, 2010

### rainwyz0706

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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!

2. Feb 22, 2010

### vela

Staff Emeritus
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.

3. Feb 23, 2010

### rainwyz0706

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

4. Feb 23, 2010