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THEOREM 6.6.2
(a) The inverse of an orthogonal matrix is orthogonal.
(b) A product of orthogonal matrices is orthogonal.
(c) If A is orthogonal, then or .
EXAMPLE 3 for an Orthogonal Matrix A
The matrix
is orthogonal since its row(and column) vectors form orthonormal sets in . We leave it for the reader to check that .
Interchanging the rows produces an orthogonal matrix for which .
Orthogonal Matrices as Linear Operators
We observed in Example 2 that the standard matrices for the basic reflection and rotation operators on and are orthogonal.
The next theorem will help explain why this is so.
THEOREM 6.6.3
If A is an matrix, then the following are equivalent.
(a) A is orthogonal
(b) for all x in .
(c) for all x and y in .
Proof We shall prove the sequence of implications .
Assume that A is orthogonal, so that . Then, from Formula 8 of Section 4.1,
