MHB JustCurious's question at Yahoo Answers (Diagonalization)

[Fernando Revilla]

Here is the question:

his problem is about diagonalization of matrices A which have n mutually orthogonal eigenvectors, each of which has length one. It is customary to write U for the matrix with columns constructed from the eigenvectors. The problem is straightforward, but requires you to follow a given suggestion. Here is the problem, followed by the suggestion.
Suppose A = UDU^-1;
where D is diagonal and U is given as above. The entries of D; U are real numbers. Show that A is equal to its transpose matrix.
Suggestion: In the diagonalization formula for A, replace U^-1 by U^T (this is valid for such matrices) and then take the transpose of both sides. Compare.
In the literature, A is called symmetric, and U is called orthogonal.

Here is a link to the question:

Diagnalization with matrices? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello JustCurious,

We have $U^{-1}=U^T$, hence $A=UDU^T$. Then, using well kown properties of transposition $$A^T=(UDU^T)^T=(U^T)^TD^TU^T=UDU^T=A$$