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Unformatted text preview: LECTURE 2 Orthogonal Vectors and Matrices OBJECTIVE: Orthogonality is central to many of the main algorithms on linear algebra. We review the ingredients: orthogonal vectors and matrices. 21 Transpose Definition The transpose A T of an m n matrix A is n m where the ( i, j ) entry of A T is the ( j, i ) entry of A : interchange the rows with the columns. Z A = 1 2 3 4 5 6 A T = 1 3 5 2 4 6 If A = A T then A is symmetric . Note The text uses A * to denote A T because it allows for complex numbers. If A = A * , then A is Hermitian. 22 Inner product Definition (Inner product) Let x , y R m . Then, the inner product of x and y is a scalar : x T y = m i =1 x i y i Definition (Euclidean length) The (Euclidean) length of a vector x is written as x : x = x T x = m i =1 x 2 i 1 2 Definition (Angle) The angle between vectors x and y satisfies cos = x T y x y 23 Some useful relationships: ( x 1 + x 2 ) T y = x T 1 y + x T 2 y x T ( y 1 + y 2 ) =...
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This note was uploaded on 03/01/2012 for the course EE 101 taught by Professor Munk during the Spring '12 term at Alaska Anch.
 Spring '12
 MUNK

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