*(Note: I'm only interested in real-valued matrices here, so I'm using "transpose" and "symmetric" instead of the more general "transjugate" and "Hermitian" in the hope that it will simplify the proof. But the theorem apparently holds for complex-valued matrices as well.)*

The Rayleigh quotient $R(M,v)$ of a symmetric matrix $M$ and a vector $v$ is defined as $\frac{v^T M v}{v^T v}$, where $x^T$ is the matrix transpose of $x$.

I've been told that the vector $v$ which gives the largest Rayleigh quotient is, in fact, the eigenvector corresponding to the largest eigenvalue of $M$. And furthermore, the value of the quotient in this case is equal to that eigenvalue. However, I've been unable to find a full proof of this fact, or an explanation of why it should work this way.

Why is there this connection between the Rayleigh quotient and the eigenvalues? Anything from an intuitive explanation to a formal proof would be appreciated.