**8**

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* : ℝ^{n} → ℝ * ^{n} is said to be* diagonalisable

The following observation is trivial but useful.

**Lemma 8.1.2** *A linear map α* : ℝ^{n} → 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.

**Definition ...**

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