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.

Homework Equations

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!

 

[vela]

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?

 

[radou]

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!