Diagonalisation for orthonormal bases
8.1 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 → ℝ n is said to be diagonalisable with respect to an orthonormal basis e1, e2,..., en If we can find λj ∈ ℝ such that α ej = λjejfor 1 ≤ j ≤ n.
The following observation is trivial but useful.
Lemma 8.1.2 A linear map α : ℝn → ℝ n is 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.