Symmetric maps

In an earlier chapter we dealt with diagonalisation with respect to some basis. Once we introduce the notion of inner product, we are more interested in diagonalisation with respect to some orthonormal basis.

Definition 8.1.1 A linear map α : ℝn → ℝnis said to be diagonalisable with respect to an orthonormal basise1, e2, …, enif we can find λj ∈ ℝ such that αej = λjejfor 1 ≤ j ≤ n.

The following observation is trivial but useful.

Lemma 8.1.2 A linear map α : ℝn → ℝnis diagonalisable with respect to an orthonormal basis if and only if we can find an orthonormal basis of eigenvectors.

Proof Left to the reader. (Compare Theorem 6.3.1.)

We need the following definitions.

Definition 8.1.3 (i) A linear map α : ℝn → ℝnis said to be symmetric if 〈αx, y〉 = 〈x, αy〉 for allx, y ∈ ℝn.

(ii) An n × n real matrix A is said to be symmetric if AT = A.

Lemma 8.1.4 (i) If the linear map α : ℝn → ℝnis symmetric, then it has a symmetric matrix with respect to any orthonormal basis.

(ii) If a linear map α : ℝn → ℝnhas a symmetric matrix with respect to some orthonormal basis, then it is symmetric.