Orthogonal Matrices and Quadratic Forms

Abstract

In Definition 1.4.2 we introduced the symmetric matrices as those square matrices that are invariant under the transpose operation. That is, those matrices A in \(\mathcal{M}_{n}(\mathbb{K})\) satisfying A = A T . This class of matrices has very important properties: for instance, they are diagonalizable (Theorem 7.4.6) and if the entries of a symmetric matrix are real, then it has only real eigenvalues (Theorem 7.4.5). Here we will study another class of matrices, whose inverses coincide with their transpose. These matrices are called the orthogonal matrices. In this section we restrict ourselves to the case of matrices with real entries. But all the results can be easily extended to the matrices with complex entries.