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Eigenvalues and Singular Values Notes

# Eigenvalues and Singular Values Notes - AIMS Lecture Notes...

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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, µ 1 3- 2 4 1 ¶ , π e 1 2- 1 . 83 √ 5- 4 7 , ( . 2- 1 . 6 . 32 ) , µ ¶ , µ 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. (Don’t forget to put a zero when a avariable doesn’t 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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Eigenvalues and Singular Values Notes - AIMS Lecture Notes...

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