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Matrix Definition: An array of numbers in m rows and n colums is called an mxn matrix a11 a12 a1n a a a 22 2n A 21 am1 am 2 amn A square matrix of order n, is an (nxn) matrix Matrix Equality Two matrices A = [aij] and B = [bij] are equal if they have the same order and aij = bij for every i and j. For example, 0.5 1 4 . 9 1 3 since both matrices 2 7 0.25 7 are of order 2x2 and all corresponding entries are equal. Matrix Addition Definition: The sum of two matrices is the matrix where each element is the sum of the corresponding elements of the two matrices. NOTE: Matrix addition is only defined when the matrices have the same dimension. Examples: 1 2 3 7 2 1 1 2 2 2 3 1 8 4 4 4 5 6 1 3 4 4 1 5 3 6 4 5 8 10 Matrix Substraction Definition: The substraction of two matrices is the matrix where each element is the difference of the corresponding elements of the two matrices. NOTE: Matrix substraction is only defined when the matrices have the same dimension. Example: 2 4 1 0 2 1 4 0 1 4 5 1 3 7 5 3 1 7 2 6 Matrix Addition Theorem 1: Let A, B, and C be matrices of the same size, then A B B A Commutative law of addition ( A B ) C A ( B C ) Associative law of addition Scalar Multiplication • Definition: Let a be a number and A=(aij)mxn be any matrix. Then aA Aa (aaij ) mxn • This means that aA is the matrix by multiplying each entry of A by the same number a 1 2 5 3 6 15 3 0 6 7 0 18 21 Negative of a Matrix • Definition: Let A=(aij) be a mxn matrix. The negative of the matrix A is the mxn matrix B=(bij) such that bij=-aij for all i,j. The negative of A is written as –A. For example, 4 0 0 1 1 4 If A 3 5 9 then A 3 5 9 7 7 2 8 2 8 Zero Matrix • Definition: The mxn matrix in which every entry is 0 is called the mxn zero matrix and is denoted by 0mxn or simply 0. • We note that A+(-A)=0, the zero matrix. • We usualy write A+(-B) as A-B Multiplication of Matrices •The product of two matrices A and B is defined if A is mxn and B is nxp •We first give the rule for multplication in special case when A is 1xn and B is nx1 b 1 Let A a1 a2 an b2 and B b n Then their product AB is defined to be a 1x1 matrix, AB a1b1 a2b2 anbn Multiplication of Matrices Now we give the rule for multiplication of matrices in general. Definition: Let A=(aij)mxn and B=(bij)nxq. Then b1 j b2 j Where cij is the product of ai1 ai 2 ain and bnj That is cij ai1b1 j ai 2b2 j ainnj AB =(cij)mxp Multiplication of Matrices 1 2 3 4 5 6 10 20 30 12 22 32 1*10+2*20+3*30 1*12+2*22+3*32 4*10+5*20+6*30 4*12+4*22+6*32 # of Columns of A must = # of Rows of B Diagonal of a Matrix Definition: If A is an nxn matrix, then the line joining (1,1) entry, (2,2) entry,…,(n,n) entry is called the diagonal (or the main diagonal) of the matrix. Definition: Any nxn matrix D is called diagonal matrix if each entry not on the diagonal is 0. For example, d1 0 0 0 d 0 D 2 0 0 dn Identity Matrix Definition: An nxn matrix A=(aij), where aij=1 whenever i=j, and aij=0 whenever i j is called the identity matrix of order n. And it is denoted by I n Square matrix with ones on the diagonal and zeros elsewhere. 1 0 I2 0 1 1 0 0 I 3 0 1 0 0 0 1 1 0 0 0 0 1 0 0 I4 0 0 1 0 0 0 0 1 It can be easily proved that if A is an mxn matrx, then I m A A AI n Properties of Matrix Multiplication 1) A( BC ) ( AB)C 2) A( B C ) AB AC 3) ( B C ) A BA CA 4) r ( AB) (rA) B A(rB ) for any scalar r 5) I m A A AI n for m n matrix A In general the followings are NOT true. AB BA If AB AC then B C If AB 0 then A 0 or B 0 Transpose Definition: Given an mxn matrix A, the transpose of A is the nxm matrix, denoted by AT, whose columns are formed from the corresponding rows of A. A (aij ) A (a ji ) T a11 a12 a1n a11 a a a a 22 2n If A 21 Then AT 12 a a a m2 mn m1 a1n a21 am1 a22 am 2 a2 n amn In other words, the ith row of AT is the ith column of A for all i. 3 1 0 Example: Let A : 1 2 4 What is AT ? Rules related to transpose: 1) ( AT )T A 2) ( A B)T AT BT 3) (rA)T rAT for any scalar r 4) ( AB)T BT AT Matrix Powers • Recall that matrix multiplication is associative, i.e. if A, B and C have the proper dimensions, then A ( BC ) ( AB ) C, so the parentheses are unnecessary and the product can be written as ABC. • If A is an n x n matrix and p is a positive integer, can define A p AA A p factors • Again, if A is an n x n matrix, adopt the convention A0 In Triangular Matrices • An n x n matrix A [ aij ] is called upper triangular if aij 0 for i > j • An n x n matrix A [ aij ] is called lower triangular if aij 0 for i < j • Note: – A diagonal matrix is both upper and lower triangular – The n x n zero matrix is both upper and lower triangular Symmetry • Defn - A matrix A is called symmetric if AT A • Defn - A matrix A is called skew-symmetric if AT A • Comment - If A is skew-symmetric, then the diagonal elements of A are zero • Comment - Any square matrix A can be written as the sum of a symmetric matrix and a skew-symmetric matrix A 1 A AT 1 A AT 2 2 symmetric skew-symmetric

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