Prove Symmetric Matrix with Orthogonal Matrix

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Homework Help Overview

The discussion revolves around proving that the product of an orthogonal matrix and a symmetric matrix results in another symmetric matrix. The original poster is exploring properties of orthogonal matrices and their relationship with symmetric matrices.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss examining the transpose of the expression (M^-1)*A*M to investigate its symmetry. There is a consideration of the properties of transposes and how they relate to symmetry.

Discussion Status

Some participants have provided hints regarding the approach to take, particularly focusing on the transpose of the matrix in question. There is an acknowledgment of the relationship between a matrix and its transpose in determining symmetry.

Contextual Notes

The original poster is working under the constraints of proving a property involving orthogonal and symmetric matrices, with an emphasis on understanding the implications of matrix transposition.

arunma
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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?
 
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It seems to me that the obvious first thing to do would be to look at the transpose of (M^-1)*A*M.
 
Hurkyl said:
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.
 
Don'cha hate when that happens!
 

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