Symmetric Matrix and Definiteness

[MikeDietrich]

Homework Statement

If A is a symmetric matrix, what can you say about the definiteness of A^2? Explain.

Homework Equations

I believe I need to use the face that A^2=SD^2S^-1.

I know that if all the eigenvalues of a symmetric matrix are positive then the matrix is positive definiteness and if the eigenvalues are positive and zero then the matrix is semidefinite.

Not sure where to go from here.

 

[micromass]

So, we need to show that A² is positive definite? What does this mean?

 

[MikeDietrich]

I don't know if A^2 will be positive definite. Oh, wait... if A^2= SD^2S^-1 then the diagonal matrix squared will only give positive or zero eigenvalues so A^2 will be positive semidefinite unless A is invertible then it would be positive definite. Do I have to worry about the orthogonal matrix and its inverse?

 

[micromass]

Yes, A² will always be positive definite, unless A is zero...