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1 Systems of Linear Algebraic Equations Part Three Definition (Band Matrix) A n×n square matrix A is called a band matrix if there exists positive integers p and q, with 1 < p and q < n such that aij = 0 for p ≤ j − i or q ≤ i − j. 02 a11 a 21 a31 A a41 a51 a61 a12 a13 a14 a15 a16 a22 a23 a24 a25 a26 a32 a33 a34 a35 a36 a42 a43 a44 a45 a46 a52 a53 a54 a55 a56 a62 a63 a64 a65 a66 p 1 p2 qn q4 aij 0 for 2 j i aij 0 for 4 i j 20 Definition (Band Matrix) The number p describes the number of diagonals above, and including, the main diagonal on which nonzero entries may lie. The number q describes the number of diagonals below, and including, the main diagonal on which nonzero entries may lie. 03 Definition (Band Matrix) The number p + q − 1 is called the bandwidth of the matrix A, which tells us how many of the diagonals can contain nonzero entries. For example, the following matrix 04 Definition (Band Matrix) the following matrix is banded with p = 3 and q = 2, and so the bandwidth is equal to 4. 05 Definition (Band Matrix) An important property of the band matrix is called the tridiagonal matrix, in this case p = q = 2, that is, all nonzero elements lie either on or directly above or below the main diagonal. 06 Definition (Band Matrix) For such type of matrix, the Gaussian elimination is particular simpler. In general, the nonzero elements of a tridiagonal matrix lie in three bands: the superdiagonal, diagonal and subdiagonal. 07 Definition (Band Matrix) For example, is a tridiagonal matrix. 1 2 2 3 1 3 2 2 A 2 4 3 1 2 3 3 4 A matrix which is predominantly zero is called a sparse matrix. A band matrix or a tridiagonal matrix is a sparse matrix but the nonzero elements of a sparse matrix are not necessarily near the diagonal. 08 The Determinant of Matrix The determinant is a certain kind of a function that associates a real number with a square matrix. We will denote the determinant of a square matrix A by det(A) or |A|. 09 Definition (Determinant of Matrix ) Let A = (aij) be an n × n square matrix then a determinant of A is given by: 1. det(A) = a11 , if n = 1. 2. det(A) = a11a22 − a12a21, if n = 2. 10 Notice that the determinant of a 2 × 2 matrix is given by the difference of the products of the two diagonals of a matrix. The determinant of a 3 × 3 matrix is defined in terms of determinants of 2 × 2 matrices and the determinant of a 4 × 4 matrix is defined in terms of determinants of 3 × 3 matrices and so on. 11 Other way to find the determinants of only 2 × 2 and 3 × 3 matrices can be found easily and quickly using diagonals (or direct evaluation). For 2 × 2 matrix, the determinant can be obtained by forming the product of the entries on the line from left to right and subtracting from this number the product of the entries on the line from right to left. 12 For a matrix of size 3×3 , the diagonals of an array consisting of the matrix with the two first columns added to the right are used. Then the determinant can be obtained by forming the sum of the products of the entries on the lines from left to right, and subtract from this number the products of the entries on the lines from right to left, as shown in Figure. 13 14 Thus for 2 × 2 matrix |A| = a11a22 − a12a21, and for 3 × 3 matrix |A| = a11a22a33 + a12a23a31 + a13a21a32 − a13a22a31 − a11a23a32 − a12a21a33 (diagonal products from left to right) (diagonal products from right to left) For finding the determinants of the higher-order matrices, we will define the following concepts of minor and cofactor of the matrices. 15 Definition (Minors of a Matrix) The minor Mij of all elements aij of a matrix A of order n×n as the determinant of the sub-matrix of order (n − 1) × (n − 1) obtained from A by deleting the ith row and jth column (also called ijth minor of A). 16 Definition (Cofactor of a Matrix) The cofactor Aij of all elements aij of a matrix A of order n × n is given by i+j Aij = (−1) Mij , where Mij is the minor of all elements aij of a matrix A. 17 Definition (Cofactor Expansion of Determinant of a Matrix) Let A be a square matrix, then we define determinant of A is the sum of the products of the elements of the first row and their cofactors. If A is 3 × 3 matrix, then its determinant is define as det(A) = |A| = a11A11 + a12A12 + a13A13. 18 Similarly, more general for n × n matrix, we define as det( A) A a A , n 2, n ij ij 1 where summation is on i for any fixed value of jth column (1 ≤ j ≤ n), or on j for any fixed value of ith row (1 ≤ i ≤ n) and Aij is the cofactor of element aij . 19 Theorem (The Laplace Expansion Theorem) The determinant of an n × n matrix A = {aij}, when n ≥ 2, can be computed as det( A) a A a A ... a A a A , which is called the cofactor expansion along the ith row and also as det( A) a A a A ... a A a A , n i1 i1 i2 i2 in in j 1 ij ij ij ij n 1j 1j 2j 2j nj nj i 1 is called cofactor expansion along jth column. It is called Laplace Expansion Theorem. 20 Note that the cofactor and minor of an element aij differs only in sign, that is, Aij = ±Mij . A quick way for determining whether to use the + or − is to use the fact that the sign relating Aij and Mij is in the ith row and jth column of the checkerboard array 21 Definition (Cofactor Matrix) If A is any n × n matrix and Aij is the cofactor of aij , then the matrix is called the matrix of cofactor from A. 22 Definition (Adjoint of a Matrix) If A is any n × n matrix and Aij is the cofactor of aij of A, then the transpose of this matrix is called the adjoint of A and is denoted by Adj(A). 23 Theorem (Properties of the Determinant) Let A be an n × n matrix: 1. The determinant of a matrix A is zero if any row or column is zero or equal to a linear combination of other rows and columns. 2. A determinant of a matrix A is changed in sign if the two rows or two columns are interchange. 24 3. The determinant of a matrix A is equal to the determinant of its transposed . 4. If the matrix B is obtained from the matrix A by multiplying every element in one row or in one column by k, then determinant of the matrix B is equal to k times the determinant of A. 25 5. If the matrix B is obtained from the matrix A by adding to a row (or a column) of a multiple of another row (or another column) of A, then determinant of the matrix B is equal to the determinant of A. 6. If two rows or two columns of a matrix A are identical, then the determinant is zero. 26 7. The determinant of a product of matrices is the product of the determinants of all matrices. 8. The determinant of a triangular matrix (uppertriangular or lower-triangular matrix) is equal to the product of all their main diagonal elements. 27 9. The determinant of an n×n matrix A times scalar multiple k equal to kn times the determinant of the matrix A, that is det(kA) = kn det(A). 10. The determinant of the kth power of a matrix A equal to the kth power of the determinant of the matrix A, that is det(Ak) = (det(A)) k. 11. The determinant of a scalar matrix (1 × 1) is equal to the element itself. 28 3 2 1 A 1 6 3 2 4 0 A13 M 11 ( 1)11 6 3 4 0 cof ( A) ( 1) 2 1 24 01 31 2 1 ( 1) 6 3 ( 1) ( 1) ( 1) 1 2 1 3 2 0 2 2 3 1 2 0 3 2 3 1 1 3 2 4 3 2 3 2 ( 1) 2 4 3 3 3 2 ( 1) 1 6 ( 1) 1 3 1 6 6 16 12 cof (A) 4 2 16 12 10 16 T cof ( A)T 20 6 16 4 12 12 12 4 2 16 adj( A) 6 2 10 12 10 16 16 16 16 Theorem If A is an invertible matrix, then 1. det(a) 0 2. det(A ) det(1A) ( A) A 4. 3. A Adj (adj( A)) adj( A det( A) det( A) 1 1 5. 29 Adj( A) det(adj( A)) det( A) 1 1 ) adj( A1 ) By using this Theorem we can find the inverse of a matrix by showing that determinant of a matrix not equal to zero and by using adjoint and determinant of the given matrix A. Matrix Inversion Method If matrix A is nonsingular, then the linear system Ax = b, always has a unique solution for each b, since the inverse matrix A−1 exists, so the solution of this system can formally expressed as A1 Ax A1b Ix A1b, gives x A1b 30 If A is a square invertible matrix, there exists a sequence of elementary row operations that carry A to the identity matrix I of the same size, that is, A → I. This same sequence of row operations carries I to A−1, that is, I → A−1. This can be also written as [A| I] → [I |A -1]. 20 Theorem For an n × n matrix A, the following properties are equivalent: 1. The inverse of matrix A exists, that is, A is nonsingular. 2. The determinant of matrix A is nonzero. 3. The homogeneous system Ax = 0 has a trivial solution x = 0. 4. The nonhomogeneous system Ax = b has a unique solution. 20 Not all matrices have inverses. Singular matrices don’t have inverse and thus the corresponding system of equations does not have a unique solution. The inverse of a matrix can also be computed by using the following numerical methods for linear systems, called, Gauss-elimination method, Gauss-Jordan method and LU-decomposition method but the best and simplest method for finding the inverse of a matrix is to perform the Gauss-Jordan method on the augmented matrix with identity matrix of same size. 20 20