- #1
3.141592654
- 85
- 0
Homework Statement
Let A be an m x n matrix with rank(A) = m < n. As far as the eigenvalues of [itex]A^{T}A[/itex] is concerned we can say that...
Homework Equations
The Attempt at a Solution
If eigenvalues exist, then
[itex]A^{T}A[/itex]x = λx where x ≠ 0.
The only thing I think I can show is that 0 is an eigenvalue:
If 0 is an eigenvalue for [itex]A^{T}A[/itex] then
[itex]A^{T}A[/itex]x = (0)x where x ≠ 0.
N(A) ≠ {0}, so Ax = 0 where x ≠ 0.
Therefore [itex]A^{T}(Ax) = 0[/itex] where x ≠ 0. So λ = 0 is an eigenvalue for [itex]A^{T}A[/itex].
Is there anything else that can be said about the eigenvalues for this matrix?