Suppose \(A=(a_{ij})\) is a real symmetric \(n\times n\) matrix with nonnegative eigenvalues. Show that \[ |a_{ij}|\le \sqrt{a_{ii}a_{jj}} \] for all distinct \(i,j\le n\).
Solution
Since A is symmetric with nonnegative eigenvalues, we may diagonalize A as \(A = U D U ^ { T }\) with \(U\) orthogonal and \(D\) diagonal with nonnegative entries, so \(A = B ^ { T } B\) for \(B ^ { T } = U D ^ { 1 / 2 }\) . Thus, A is a Gram matrix, i.e., \(a _ { i j } = \langle v _ { i } , v _ { j } \rangle\) where \(v _ { i }\) are the columns of \(B ,\) so by the Cauchy–Schwarz inequality \(\abs{ a _ { i j } } \leq \| v _ { i } \| \| v _ { j } \| = \sqrt { a _ { i i } a _ { j j } }\) , as desired.