Proving Hv = 0 for a Symmetric Matrice with Orthogonal Diagonalization

[moo5003]

Homework Statement

Suppose H is an n by n real symmetric matrix. v is a real column n-vector and H^(k+1)v = 0. Prove that Hv = 0

The Attempt at a Solution

Since H is a real symmetric matrice we can find an orthogonal matrix Q to diagnolize it:

M = Q transpose.

MA^(k+1)Qv = 0

Implying

A^(k+1)Qv = 0

This is where I'm stuck I'm not sure how to proceed.

I'm pretty sure its not possible to somehow get Q remove from the equation because that implies v or A would have to be 0 but this does not follow since I can easily cook up an example were there is a symmetric matrix to a power were H^(K+1)v=0 and v != 0. Thus any hints would be appreciated.

 

[Dick]

If you work in the basis where H is diagonal then replace the phrase "n by n real symmetric matrix" in your premise with "n by n DIAGONAL matrix". Can you prove that one?

 

[StatusX]

Assume v is not zero. Then, since Q is invertible, Qv is not zero. So Qv must be in the nullspace of A^(k+1). What is the null space of a diagonal matrix, and how is this related to the nullspace of its powers?

 

[moo5003]

If you work in the basis where H is diagonal then replace the phrase "n by n real symmetric matrix" in your premise with "n by n DIAGONAL matrix". Can you prove that one?

That was the proof shown in class where you chose to represent the matrix using a basis consisting of eigen vectors, though I didnt start my proof that way and I believe that the diagonalization method should work if I could finish the last step.

Statusx: It would imply that Qv is in the nullspace of A itself since A^(k+1) simply exponentiates each entry by k+1 (diagonal matrice), though I'm unsure how that shows Av = 0.

You would get AQv = 0

Transpose of Qv (easier as a row vector when typing) would be something along the lines:

[0,0,0,...,a_i,...,0] where a_i can by any real number corresponding to a 0 in the diagonal entry a_ii in the diagonal matrice A.

 

[StatusX]

You would get AQv = 0

Right, and so MAQv = Hv = 0.