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Unformatted text preview: 6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I . Note. We will see that the columns of an orthogonal matrix must be unit vectors and that the columns of an orthogonal matrix are mutually orthogonal (inspiring a desire to call them orthonormal matrices , but this is not standard terminology). Theorem 6.5. Characterizing Properties of an Orthogonal Matrix. Let A be an n n matrix. The following conditions are equivalent: 1. The rows of A form an orthonormal basis for R n . 2. The columns of A form an orthonormal basis for R n . 3. The matrix A is orthogonal that is, A is invertible and A 1 = A T . 6.3 Orthogonal Matrices 2 Proof. Suppose the columns of A are vectors ~a 1 , ~a 2 , . . . , ~a n . Then A is orthogonal if and only if I = A T A = ~a 1 ~a 2 . . . ~a n . . . . . . . . . ~a 1 ~a 2 ~a n . . . . . . . . . and we see that the diagonal entries of the product are ~a j ~a j = 1 therefore each vector is a unit vector. All off-diagonal entries of I are 0 and so for i 6 = j we have ~a i ~a j = 0. Therefore the columns of A are orthonormal...
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