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6.6 In this section we shall develop properties of square matrices with orthonormal
column vectors. Such matrices arise in many contexts, including problems
ORTHOGONAL MATRICES involving a change from one orthonormal basis to another.
Matrices whose inverses can be obtained by transposition are sufficiently important that there is some terminology associated with
them.
DEFINITION
A square matrix A with the property
is said to be an orthogonal matrix.
It follows from this definition that a square matrix A is orthogonal if and only if
(1)
In fact, it follows from Theorem 1.6.3 that a square matrix A is orthogonal if either or .
EXAMPLE 1 A Orthogonal Matrix
The matrix
is orthogonal, since
EXAMPLE 2 A Rotation Matrix Is Orthogonal
Recall from Table 6 of Section 4.2 that the standard matrix for the counterclockwise rotation of through an angle is
