| # Rayleigh quotients Theorem Fo a symmetric matrix , we can express the smallest and largest eigenvalues, and , as Proof: The proof of the expression above derives from the SED of the matrix, and the invariance of the Euclidean norm constraint under orthogonal transformations. We show this only for the largest eigenvalue; the proof for the expression for the smallest eigenvalue follows similar lines. Indeed, with , we have Now we can define the new variable , so that , and express the problem as Clearly, the maximum is less than . That upper bound is attained, with for an index such that , and for . This proves the result. This corresponds to setting , where is the eigenvector corresponding to .