-
Let
be a self-adjoint matrix. The Rayleigh quotient for is defined as the scalar -
(Friedberg 6.36) For a self adjoint matrix
, is the largest eigenvalue of . Similarly, is the smallest eigenvalue of . -
Proof. Choose an orthonormal basis consisting of eigenvectors of
. Represent Assume that the eigenvalues are sorted
. Computing
, we have We can clearly bound this as
-