Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
148094864 Chapter 1 Matrices 1.1 Basic Properties of Matrices A matrix A is a rectangular array of elements denoted by a11 a 21 A= a m1 a12 a 22 am 2 a1n ... a 2 n ... a mn ... (1.1-1) This matrix A is called an mn matrix or a matrix of order “m by n” referring to the fact that A has m rows and n columns. For a square matrix m is equal to n. Element aij is located in the ith row and jth column. (1) Matrix equality: A = B if they are of the same order and aij = bij (2) Column matrix [m, 1] has m rows and one column. 1 A[3, 1] = 3 2 (3) Row matrix [1, n] has 1 rows and n column. A[3, 1] = 1 3 2 (4) Transposition: When a matrix is transposed, the rows and column are interchanged. 1 2 3 T A= A = 4 5 6 1 4 2 5 ; (a )T = a ij ji 3 6 (5) Symmetric matrix: symmetric matrix is one that is equal to its own transpose. 1 1 2 A = A or aij = aji. Example: A = 1 5 4 4 6 2 T 1 148094864 1 1 2 (6a) Skew matrix: aij = aji; and not all aii = 0. Example: A = 1 5 4 2 4 6 0 1 2 (6b) Skew-symmetric matrix: aij = aji; and aii = 0. Example: A = 1 0 4 2 4 0 (7) Complex conjugate of a matrix A is a matrix derived from A by taking the complex conjugate of all the elements in A. 2 3 i 2 3 i A= A = 2i 1 i 2i 1 i (8) Hermitian matrix is equal to the transpose of its own complex conjugate. 3 i 3 i 4 4 A= A = 2 2 3 i 3 i 3 i 4 T A* = A = =A 2 3 i (9) Skew-Hermitian matrix is equal to negative of the transpose of its own complex conjugate. 2 i 2 i 2 i 0 0 0 T A= = = A A 2 i 2 i 0 0 0 2 i 2 i 0 T A = A = 0 2 i (10) Zero or null matrix has all its elements zero and is denoted by 0, or, aij = 0 for all i and j. (11) Identity or unit matrix I has the following properties aij = ij = 1 if i = j aij = ij = 0 if i j (12) Matrix addition and subtraction is defined only for matrices of the same order. C = A + B, cij = aij + bij 2 148094864 D = A B, dij = aij bij (13) Matrix multiplication by a scalar C = kA cij = kaij (14) Matrix multiplication A[m, p] B[p, n] = C[m, n] A is said to be “post-multiplied” by B, or B is “pre-multiplied” by A, to distinguish the order in which A and B appear. The element cij is obtained from the multiplication of the row i of matrix A and the column j of matrix B. p cij = a k 1 b ik kj 1 2 3 Example: A = , B = 4 5 6 1 1 1 1 2 0 3 1 0 1 2 3 A[2, 3] B[3, 4] = C[2, 4] 3 c11 = a k 1 b = a11b11 + a12b21 + a13b31 = (1)(1) + (2)(2) + (3)(0) = 5 1k k 1 3 c23 = a k 1 Similarly, b = a21b13 + a22b23 + a23b33 = (4)(1) + (5)(3) + (6)(2) = 31 2k k 3 5 2 13 10 C= 14 2 31 19 (15) Matrix partition a11 a 21 A= a 31 a 41 a12 a13 a14 a 22 a 23 a 24 a 32 a 33 a 34 a 42 a 43 a 44 a11 a12 a a 22 If A11 = 21 a31 a32 a13 a 23 a33 a15 a 25 a 35 a 45 , A12 = 3 a14 a 24 a34 a15 a 25 a35 148094864 A21 = a 41 A Then A = 11 A21 a 42 a 43 , A22 = A11 T A12 T and A = T A22 A12 a 44 a 45 A21 T A22 T The following example shows the use of Matlab commands for matrix partition. Matlab Example __________________________________________________ Example 1.1 >> A=[1 2 3 4 5;2 -1 2 5 6;3 2 -3 4 1;4 -1 -2 1 1] A= 1 2 3 4 5 2 -1 2 5 6 3 2 -3 4 1 4 -1 -2 1 1 >> A11=A(1:3,1:3) A11 = 1 2 3 2 -1 2 3 2 -3 >> A12=A(1:3,4:5) A12 = 4 5 5 6 4 1 5 >> A21=A(4,1:3) A21 = 4 -1 -2 >> A22=A(4,4:5) A22 = 1 1 >> At=A' At = 1 2 2 -1 3 2 4 5 5 6 3 2 -3 4 1 4 -1 -2 1 1 4 148094864 >> At=[A11' A21';A12' A22'] At = 1 2 3 4 2 -1 2 -1 3 2 -3 -2 4 5 4 1 5 6 1 1 A set of n simultaneous linear equations with n unknowns can be written in matrix form as Ax = b where A is the coefficient matrix, x is the column vector of the unknowns, and b is the column vector of the right hand side. For a set of two equations and two unknowns a11x1 + a12x2 = b1 a21x1 + a22x2 = b2 a12 a A = 11 , x = a a 21 22 x1 x , and b = 2 b1 b 2 5