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LECTURE8: BASICMATRIXALGEBRA Prof. N. Harnew University of Oxford MT2012 1 Outline: 8. BASIC MATRIX ALGEBRA 8.1 Addition of matrices 8.1.1 Properties and an example 8.2 Multiplication by a scalar 8.2.1 Example 8.3 Multiplication of matrices 8.3.1 Example 1 8.3.2 Example 2 8.4 Properties of matrix multiplication 8.4.1 Proof of A(BC) = (AB)C 2 8.1 Addition of matrices ◮ Matrix summation follows the properties of linear algebra C=A+B Hence C =(A+B) =A +B ij ij ij ij ◮ Writing this out: A A · · · A B B · · · B 11 12 1n 11 12 1n A A · · · A B B · · · B C=A+B= 21 22 2n + 21 22 2n ··· · · · · · · · · · ··· · · · · · · · · · A A · · · A B B · · · B m1 m2 mn m1 m2 mn (1) A11+B11 A12 +B12 · · · A1n +B1n = A21+B21 A22 +B21 · · · A2n +B2n (2) · · · · · · · · · · · · Am1 +Bm1 Am2 +Bm1 · · · Amn +Bmn ◮ For this to have any meaning, both matrices must have the same dimensions (in this case m × n). 3 8.1.1 Properties and an example ◮ It follows obviously (and also from the rules of linear operators) that C=A+B=B+A(commutative). ◮ The difference of two matrices follows in an obvious way: C=A−BhaselementsC =A −B . ij ij ij Example C=A+B= 1 2 3 + 7 8 9 = 8 10 12 4 5 6 −2 −1 0 2 4 6(3) 4
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