Lecture9 - 1 Lecture 9 9.1 Orthogonal matrices A matrix A...

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Unformatted text preview: 1 Lecture 9 9.1 Orthogonal matrices A matrix A is orthogonal if 1) A is n n matrix 2) A T A = I. Let nn n n n n a a a a a a a a a A 2 1 2 22 21 1 12 11 . n a a a 2 1 Characterizing properties of an orthogonal n n matrix A: a) The rows of A form an orthonormal basis for R n . b) The columns of A form an orthonormal basis for R n . c) A-1 = A T . d) A-1 , A T orthogonal. e) det(A) = 1 f) If B is orthogonal, then AB is also orthogonal. Example: Verify that the matrix 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 A is an orthogonal matrix, and find A-1 . 2 Thm. Let A be an orthogonal n n matrix and let x and y be any column vectors in R n . 1) y x y A x A ) ( ) ( 2) || || || || x x A 3) The angle between x and y = The angle between x A and y A , for nonzero vectors. Examples: Let 4 2 1 , 3 , , q q q q B be an orthonormal basis of R 4 and v be a vector in R 4 . If B v = [3, -4, -3, 2], find || v ||....
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Lecture9 - 1 Lecture 9 9.1 Orthogonal matrices A matrix A...

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