lec2 - McGill University Math 270: Applied Linear Algebra...

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Unformatted text preview: McGill University Math 270: Applied Linear Algebra CHAPTER 2: MATRICES AND DETERMINANTS 1 Matrix Algebra Suppose given two matrices A = [ a ij ] and B = [ b ij ] . The two matrices are equal, A = B, if and only if They have the same dimensions; All corresponding elements are equal: a ij = b ij for all ( i,j ) . 1.1 Matrix Addition Addition (or subtraction) of A and B with the same dimension gives the matrix C = [ c ij ] = A B : c ij = ( A B ) ij = a ij b ij for all ( i,j ) . 0-0 1.2 Scalar Multiplication of Matrix Multiplication of a matrix by a scalar, C = [ c ij ] = A : c ij = ( A ) ij = a ij for all ( i,j ) . 1.3 Matrix Multiplication Given A = [ a ij ] m n , B = [ b ij ] n p . The multiplication of A and B is defined as C = [ c ij ] n p = AB : c ij = ( AB ) ij = n X k =1 a ik b kj ( i = 1 , ,m ); ( j = 1 , ,p ) . From the above, we derive 1. Given matrix A = [ a ] m n , and a column n- vector b = [ b i ] n , which implies p = 1 . then we have ( A b ) i = n X k =1 a ik b k ( i = 1 , ,m ); 2. Given matrix A = [ a ] m n , and B = [ b 1 , b 2 , , b p ] , then we have AB = [ A b 1 ,A b 2 , A b p ] 0-1 1.4 Some Properties of Matrix Multiplication Suppose that matrices A,B,C have appropriate dimensions for the operations to be performed. Then, one has: Matrix multiplication is associative and distributive over addition: A ( BC ) = ( AB ) C A ( B + C ) = AB + AC ( A + B ) C = AC + BC Matrix multiplication is not commutative: AB 6 = BA. 2 Some Important ( n n ) Matrices Suppose we have the matrix, A = [ a ij ] n n . A is the zero matrix , if a ij = 0 , for i = 1 , ,n ; j = 1 , ,n. A is the identity matrix , A = I n if a ij = ij = 1 , ( i = j ) , ( i 6 = j ) . It follows that I n I n = I n . 0-2 A is the upper (or lower) triangular matrices , if a ij = 0 , for i > j, (or i < j ) . A is the diagonal matrices , if a ij = 0 , for i 6 = j. Namely, A = D = d 11 d 22 . . . d nn It is derived D k = d k 11 d k 22 . . . d k nn A is the symmetric matrices , if A is square ( m = n ) , and a ij = a ji . A is the antisymmetric matrices or skew symmetric , if A is square ( m = n ) , and a ij =- a ji . 0-3 3 Partitioning of Matrices Any matrix A may be partitioned into a number of blocks by vertical lines that extend from bottom to top and horizontal lines extend from left to right. For instance, we can write A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = [ A 11 A 12 A 13 . ] Note: The usual matrix arithmetic can be carried out with partitioned matrices. For instance, if A = A 11 A 12 A 1 n ....
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lec2 - McGill University Math 270: Applied Linear Algebra...

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