Answer to Question #194356 in Linear Algebra for Kentse Modisane

Question

 Let A be any n x n matrix and let P be an n x n orthogonal matrix. Prove that jAj D jP

APj:

Answer 1

A is $n\times n$ symmetric matrix.

its all eigen values are greater than 0.

Characteristics polynomial is $det(A-\lambda I_n)=0$

$\lambda$ are the eigenvalues of A.

As P is orthogonal matrix so $P^T=P^{-1}$

Characteristics polynomial for $P^TAP$ is given as-

$det(P^TAP-\lambda I_n) \\ =det(P^{-1}AP-\lambda P^{-1}P) \\ =det{P^{-1}(A-\lambda I_n)P} \\ =det(P^{-1})det (P)det(A-\lambda I_n) \\ =det(P^{-1}P)det(A-\lambda I_n) \\ =det(A-\lambda I_n)$

$(P^TAP)^T=(PA)^T(P^T)^T=P^TA^TP=P^TAP, as A^T=A$

So $P^TAP$ is symmetric matrix.

From characteristics equation, eigenvalues of $P^TAP$ is the same as eigenvalues of A.