finding eigenvalues by optimization

If you find \(\arg\max_{x} ||Ax||_2\) subject to \(||x||=1\), then you will get the largest eigenvalue of \(A\).

For the case where \(A\) is symmetric, this can be seen because \(A\) can be diagonalized by a unitary matrix so \(A=PDP^T\). Then, imagine multiplying any vector \(x\) by \(PDP^T\). Both \(P\) and \(P^T\) are unitary, so they will not change the norm of the vector. So the way to get the biggest vector is to make sure that \(y=P^Tx\) puts all its weight, i.e. is a one hot vector, such that \(y\) can select the biggest number on the diagonal of \(D\).