Tags: linear algebra, quiz-03, spectral theorem, eigenvectors, diagonalization, lecture-04
Suppose \(A\) is a \(d \times d\) symmetric matrix.
True or False: There exists an orthonormal basis in which \(A\) is diagonal.
True.
By the spectral theorem, every \(d \times d\) symmetric matrix has \(d\) mutually orthogonal eigenvectors. If we normalize these eigenvectors, they form an orthonormal basis.
In this eigenbasis, the matrix \(A\) is diagonal: the diagonal entries are the eigenvalues of \(A\).