# ch2 - McGill University Math 270 Applied Linear Algebra...

This preview shows pages 1–6. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ] = αA : c ij = ( αA ) ij = αa 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 ∑ 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 ∑ 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 ̸ = 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 = δ ij = { 1 , ( i = j ) , ( i ̸ = 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 ̸ = 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 2.1 Row Elementary Operation Matrices Recall that in last chapter, we have defined three types of ele- mentary row operations. By allying these operations, one that transform the ( n × n ) matrix ( A ) into ( ˜ A ) . Specifically, • By (EOP)-1: R i ⇐⇒ R j , ( A ) ⇐⇒ ( ˜ A ) 1 ; • By (EOP)-2: R i ⇐⇒ kR i , ( A ) ⇐⇒ ( ˜ A ) 2 ; • By (EOP)-3: R i ⇐⇒ R i + kR j , ( A ) ⇐⇒ ( ˜ A ) 3 ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 36

ch2 - McGill University Math 270 Applied Linear Algebra...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online