Lecture+6-Linear+Algebra+_+MATLAB+graphics

Lecture+6-Linear+Algebra+_+MATLAB+graphics - Matrix...

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Matrix Multiplications Recall how matrix multiplication works g j a b c ag bh ci aj bk cl h k d e f dg eh fi dj ek fl i l g j ga jd gb je gc jf a b c h k ha kd hb ke hc kf d e f i l ia ld ib le ic lf + + + + = + + + + + + + = + + + + + + [A]*[B] A 2 x 3 B 3 x 2 C 2 x 2 =A 2 x 3 *B 3 x 2 B 3 x 2 A 2 x 3 D 3 x 3 =B 3 x 2 *A 2 x 3
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Matrix multiplication can be performed only if the inner dimensions are equal Matrix Multiplication [A] m x n [B] n x l [C] m x l
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Interior dimensions have to be equal For a vector We will be using square matrices (m=n) j i i j k k A B C × × × × = m n n m A x b × × = n n n n b x A = × Matrix Multiplication x n x 1
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MATLAB In Fortran, the matrix multiplication have to be done by Do Loops In MATLAB, it is automatic A*B = C Note no period (not element-by-element operation) For vectors A*x = b
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Matrix Multiplication Associative ( [ A ] [ B ] ) [ C ] = [ A ] ( [ B ] [ C ] ) Distributive [ A ] ( [ B ] + [ C ] ) = [ A ] [ B ] + [ A ] [ C ] ([ A ] + [ B ] ) [ C ] = [ A ] [ C ] + [ B ] [ C ] Not generally commutative [ A ] [ B ] [ B ] [ A ]
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Matrix Inverse Matrix division is undefined However, there is a matrix inverse for non-singular square matrices [ A ] - 1 [ A ] = [ A ] [ A ] - 1 = [ I ] Matlab: >>inv(A) Multiplication of a matrix by the inverse is analogous to division
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11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 a a a a a a a a A a
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Lecture+6-Linear+Algebra+_+MATLAB+graphics - Matrix...

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