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la10-om - LINEAR ALGEBRA W W L CHEN c W W L Chen 1997 2005...

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LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2005. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 10 ORTHOGONAL MATRICES 10.1. Introduction Definition. A square matrix A with real entries and satisfying the condition A 1 = A t is called an orthogonal matrix. Example 10.1.1. Consider the euclidean space R 2 with the euclidean inner product. The vectors u 1 = (1 , 0) and u 2 = (0 , 1) form an orthonormal basis B = { u 1 , u 2 } . Let us now rotate u 1 and u 2 anticlockwise by an angle θ to obtain v 1 = (cos θ, sin θ ) and v 2 = ( sin θ, cos θ ). Then C = { v 1 , v 2 } is also an orthonormal basis. θ u 1 u 2 v 1 v 2 Chapter 10 : Orthogonal Matrices page 1 of 10
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Linear Algebra c W W L Chen, 1997, 2005 The transition matrix from the basis C to the basis B is given by P = ([ v 1 ] B [ v 2 ] B ) = cos θ sin θ sin θ cos θ . Clearly P 1 = P t = cos θ sin θ sin θ cos θ . In fact, our example is a special case of the following general result. PROPOSITION 10A. Suppose that B = { u 1 , . . . , u n } and C = { v 1 , . . . , v n } are two orthonormal bases of a real inner product space V . Then the transition matrix P from the basis C to the basis B is an orthogonal matrix. Example 10.1.2. The matrix A = 1 / 3 2 / 3 2 / 3 2 / 3 1 / 3 2 / 3 2 / 3 2 / 3 1 / 3 is orthogonal, since A t A = 1 / 3 2 / 3 2 / 3 2 / 3 1 / 3 2 / 3 2 / 3 2 / 3 1 / 3 1 / 3 2 / 3 2 / 3 2 / 3 1 / 3 2 / 3 2 / 3 2 / 3 1 / 3 = 1 0 0 0 1 0 0 0 1 . Note also that the row vectors of A , namely (1 / 3 , 2 / 3 , 2 / 3), (2 / 3 , 1 / 3 , 2 / 3) and (2 / 3 , 2 / 3 , 1 / 3) are orthonormal. So are the column vectors of A . In fact, our last observation is not a coincidence. PROPOSITION 10B. Suppose that A is an n × n matrix with real entries. Then (a) A is orthogonal if and only if the row vectors of A form an orthonormal basis of R n under the euclidean inner product; and (b) A is orthogonal if and only if the column vectors of A form an orthonormal basis of R n under the euclidean inner product. Proof. We shall only prove (a), since the proof of (b) is almost identical. Let r 1 , . . . , r n denote the row vectors of A . Then AA t = r 1 · r 1 . . . r 1 · r n . . . . . . r n · r 1 . . . r n · r n . It follows that AA t = I if and only if for every i, j = 1 , . . . , n , we have r i · r j = 1 if i = j , 0 if i = j , if and only if r 1 , . . . , r n are orthonormal. PROPOSITION 10C. Suppose that A is an n × n matrix with real entries. Suppose further that the inner product in R n is the euclidean inner product. Then the following are equivalent: (a) A is orthogonal.
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