By using the definition of eigenvalue and the property of orthogonal matrix, It is easy to prove that L is the eigenvalue of the matrix P(transposed)AP when P is an orthogonal matrix(note that P(transposed)P=E).

the corresponding eigenvector of P(transposed)AP is P(transposed)X.

If P(transposed)AP is a diagonal matrix D,then D gives all the eigenvalue of A, and the corresponding eigenvector of each eigenvalue coincide with the corresponding column vectors of P.