What Are the Eigenvalues of A Transpose A?

[3.141592654]

Homework Statement

Let A be an m x n matrix with rank(A) = m < n. As far as the eigenvalues of A^{T}A is concerned we can say that...

Homework Equations

The Attempt at a Solution

If eigenvalues exist, then

A^{T}Ax = λx where x ≠ 0.

The only thing I think I can show is that 0 is an eigenvalue:

If 0 is an eigenvalue for A^{T}A then

A^{T}Ax = (0)x where x ≠ 0.

N(A) ≠ {0}, so Ax = 0 where x ≠ 0.

Therefore A^{T}(Ax) = 0 where x ≠ 0. So λ = 0 is an eigenvalue for A^{T}A.

Is there anything else that can be said about the eigenvalues for this matrix?

 

[I like Serena]

Did you hear about the spectral theorem for symmetric matrices?

 

[3.141592654]

No, that wasn't covered in my course so I suppose that's not what the professor is looking for. Is it relatively ea

 

[Dick]

No, that wasn't covered in my course so I suppose that's not what the professor is looking for. Is it relatively ea

If A^T*A*x=lambda*x what happens if you multiply both sides on the left by x^T? No, you don't need the spectral theorem.

 

[I like Serena]

No, that wasn't covered in my course so I suppose that's not what the professor is looking for. Is it relatively ea

Spectral theorem says that a symmetric matrix is diagonalizable.
In particular, a real nxn symmetric matrix has n real eigenvalues.