Question

4. Prove that the eigenvalues of A and AT are identical. 5. Prove that the eigenvalues of a diagonal matrix are equal to the diagonal elements. 6. Consider the matrix ompute the eigenvalues and eigenvectors of A, A-,

Linear Algebra

Answer 1

Let A be a square matrix. Recall that the eigen values of a matrix are roots of its characteristic polynomial. Hence if the matrices A and A^T have the same characteristic polynomial, then they have the same eigen values. So we show that the characteristic polynomial P_A (t) = det (A - t I) of A is the same as the characteristic polynomial P_A^T (t) = det (A^T - t I) of the transpose A^T. We have P_A^T 9t) = det (A^T - t I) = det (A^T - t I^T) Because I^T = I = det ((A - t I)^T) det (B^T) = det (B) \r\n = det ((A - t I)) = P_A (t) Therefore We obtain P_A^T (t) = P_A (t), and we conclude that the eigen values of A and A^T are the same (Identical). Diagonal matrix is a square (n times n) matrix with non-zero entries only along the diagonal from the under left