Suppose that matrix A is real positive-definite and symmetric. Then there is always a matrix T=BW such that $T^T A T$ is the identity matrix. W is the diagonal matrix of the reciprocals of the square roots of eigen values of A (i.e. $W_{i,i}=1/\sqrt{\lambda_i}$ and B is the matrix that contains the corresponding eigenvectors of A as columns.
Also, some related question is, if we have $\phi$ being the matrix that contains the eigenvectors of a symmetric (possibly complex) matrix C. Can we be sure that $\phi^T\phi$ is always diagonal? or does this only hold for real matrix C
Thanks in advance