Linear Algebra: Concepts and Methods

In this chapter, we look at orthogonal diagonalisation, a special form of diagonalisation for real symmetric matrices. This has some useful applications: to quadratic forms, in particular.

11.1 Orthogonal diagonalisation of symmetric matrices

Recall that a square matrix A = (ai j) is symmetric if AT = A. Equivalently, A is symmetric if ai j = aji for all i, j ; that is, if the entries in opposite positions relative to the main diagonal are equal. It turns out that symmetric matrices are always diagonalisable. They are, furthermore, diagonalisable in a special way.

11.1.1 Orthogonal diagonalisation

We knowwhat itmeans to diagonalise a square matrix A. Itmeans to find an invertible matrix P and a diagonal matrix D such that P-1 A P = D. If, in addition,we can find an orthogonal matrix P which diagonalises A, so that P-1 AP = PT A P = D, then this is orthogonal diagonalisation.

Definition 11.1 A matrix A is said to be orthogonally diagonalisable if there is an orthogonal matrix P such that PT AP = D where D is a diagonal matrix.

As P is orthogonal, PT = P-1, so PT A P = P-1 A P = D. The fact that A is diagonalisable means that the columns of P are a basis of ℝn consisting of eigenvectors of A (Theorem 8.22). The fact that A is orthogonally diagonalisable means that the columns of P are an orthonormal basis of ℝn consisting of an orthonormal set of eigenvectors of A (Theorem 10.21).