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Unformatted text preview: AIMS Lecture Notes 2006 Peter J. Olver 3. Review of Matrix Algebra Vectors and matrices are essential for modern analysis of systems of equations algebrai, differential, functional, etc. In this part, we will review the most basic facts of matrix arithmetic. See [ 38 ] for full details. 3.1. Matrices and Vectors. A matrix is a rectangular array of numbers. Thus, 1 3- 2 4 1 , e 1 2- 1 . 83 5- 4 7 , ( . 2- 1 . 6 . 32 ) , , 1 3- 2 5 , are all examples of matrices. We use the notation A = a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 2 . . . a mn (3 . 1) for a general matrix of size m n (read m by n ), where m denotes the number of rows in A and n denotes the number of columns . Thus, the preceding examples of matrices have respective sizes 2 3, 4 2, 1 3, 2 1, and 2 2. A matrix is square if m = n , i.e., it has the same number of rows as columns. A column vector is a m 1 matrix, while a row vector is a 1 n matrix. As we shall see, column vectors are by far the more important of the two, and the term vector without qualification will always mean column vector. A 1 1 matrix, which has but a single entry, is both a row and a column vector. The number that lies in the i th row and the j th column of A is called the ( i, j ) entry of A , and is denoted by a ij . The row index always appears first and the column index second. Two matrices are equal, A = B , if and only if they have the same size, and all their entries are the same: a ij = b ij for i = 1 , . . . , m and j = 1 , . . . , n . 3/15/06 35 c 2006 Peter J. Olver A general linear system of m equations in n unknowns will take the form a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 , a 21 x 1 + a 22 x 2 + + a 2 n x n = b 2 , . . . . . . . . . a m 1 x 1 + a m 2 x 2 + + a mn x n = b m . (3 . 2) As such, it is composed of three basic ingredients: the m n coefficient matrix A , with entries a ij as in (3.1), the column vector x = x 1 x 2 . . . x n containing the unknowns , and the column vector b = b 1 b 2 . . . b m containing right hand sides . As an example, consider the linear system x + 2 y + z = 2 , 2 y + z = 7 , x + y + 4 z = 3 , The coefficient matrix A = 1 2 1 2 1 1 1 4 can be filled in, entry by entry, from the coef- ficients of the variables appearing in the equations. (Dont forget to put a zero when a avariable doesnt appear in an equation!) The vector x = x y z lists the variables, while the entries of b = 2 7 3 are the right hand sides of the equations....
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