Let $A$ be a symmetric matrix and call $\mathcal{V}$ a subspace spanned by the columns of the orthonormal matrix $V$. I was wondering if there exists theorems that gives me the eigenvalues/eigenvectors of $V^T A V$ in function of the eigenvalues/eigenvectors of $A$. I know already that the eigenvalues of $V^T A V$ are bounded by the eigenvalues by the eigenvalues of $A$ (Cauchy interlace theorem). Can someone help me with that?
many thanks
$ V^T A V $ is known as a unitary similarity transformation of $ A $.
If $ A x = \lambda x $ then you can multiply both sides by $ V^T $ to get $ V^T A x = \lambda V^T x$. Next, insert the identity (since $V V^T = I $) $ V^T A V V^T x = \lambda V^T x $. Then place some parentheses strategically: $ (V^T A V )(V^T x ) = \lambda ( V^T x ) $ and the answer stares you right in the face.
Notice that there is no requirement that $ A $ be symmetric for this to work.