Question 4.2.1: Show that K is a symmetric matrix if K and M are symmetric. ......

To show that a matrix is symmetric, use the rule that for any two square matrices of the same size (A B)^T=B^T A^T. Applying this rule twice yields

\tilde{K}^T=\left(M^{-1 / 2} K M^{-1 / 2}\right)^T=\left(K M^{-1 / 2}\right)^T M^{-1 / 2}=M^{-1 / 2} K^T M^{-1 / 2}=M^{-1 / 2} K M^{-1 / 2}=\tilde{K}

Thus \tilde{K}=\tilde{K}^T and is a symmetric matrix.