Prove Symmetric Matrix with Orthogonal Matrix

[arunma]

I've got a question regarding orthogonal matrices. I am given an orthogonal matrix M, and a symmetric matrix A. I need to prove that (M^-1)*A*M is also symmetric (all of the matrices are n x n). I know that for an orthogonal matrix, its inverse is equal to its transpose. Can anyone give me some hints on how to begin this proof?

 

[Hurkyl]

It seems to me that the obvious first thing to do would be to look at the transpose of (M^-1)*A*M.

 

[arunma]

It seems to me that the obvious first thing to do would be to look at the transpose of (M^-1)*A*M.

Thanks for the tip. I believe the transpose of that matrix would also be (M^-1)*A*M, since (AB)^T = B^T * A^T. And if a matrix equals its own transpose, doesn't that make it symmetric?

Well, I guess that in my commentary, I've accidentally solved the problem. Thanks.

 

[HallsofIvy]

Don'cha hate when that happens!