This closing note of Topic 3 studies the second major class of normal operator: those that preserve length exactly. Called orthogonal on a real inner-product space and unitary on a complex one, these operators are already familiar in disguise — the in every factorisation (Theorem 1 of the QR-via-Gram-Schmidt note) is exactly an orthogonal operator, and this note supplies the abstract characterisation that explains why was always the right condition to check.
Learning objectives. By the end of this note you should be able to:
- state the equivalent characterisations of a unitary/orthogonal operator;
- prove that such operators are automatically normal;
- verify a small matrix is orthogonal;
- explain the eigenvalue constraint these operators satisfy.
In this lesson
- Definition and equivalent characterisations
- Eigenvalue constraint
Before you start
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