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lecture6

# lecture6 - Lecture 6 Matrix Operations and Gaussian...

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Lecture 6 Matrix Operations and Gaussian Elimination for Solving Linear Systems Shang-Hua Teng

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Matrix (Uniform Representation for Any Dimension) An m by n matrix is a rectangular table of mn numbers j i n m m m n n a j i A a a a a a a a a a A , , 2 , 1 , , 2 2 , 2 1 , 2 , 1 2 , 1 1 , 1 ) , ( write we Sometime ... ... ... ... = =
Matrix (Uniform Representation for Any Dimension) Can be viewed as m row vectors in n dimensions = n m m m n n a a a a a a a a a A , 2 , 1 , , 2 2 , 2 1 , 2 , 1 2 , 1 1 , 1 ... ... ... ...

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Matrix (Uniform Representation for Any Dimension) Or can be viewed as n column vectors in m dimensions = n m m m n n a a a a a a a a a A , 2 , 1 , , 2 2 , 2 1 , 2 , 1 2 , 1 1 , 1 ... ... ... ...
Squared Matrix An n by n matrix is a squared table of n 2 numbers = n n n n n n a a a a a a a a a A , 2 , 1 , , 2 2 , 2 1 , 2 , 1 2 , 1 1 , 1 ... ... ... ...

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Some Special Squared Matrices All zeros matrix = 0 ... 0 0 ... 0 ... 0 0 0 ... 0 0 ) , ( 0 m n Identity matrix = = 1 ... 0 0 ... 0 ... 1 0 0 ... 0 1 ) , ( n n I I
Matrix Operations Addition Scalar multiplication Multiplication

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1. Matrix Addition: + + + + + + + + = + = = mn mn m m n n n n mn m m n n mn m m n n b a b a b a b a b a b a b a b a B A b b b b b b b b b B a a a a a a a a a A , , 1 1 2 2 22 22 21 21 1 1 12 12 11 11 2 1 2 22 21 1 12 11 2 1 2 22 21 1 12 11 Matrices have to have the same dimensions What is the complexity?
2. Scalar Multiplication: = = mn m m n n mn m m n n a a a a a a a a a a a a a a a a a a A λ λ λ λ λ λ λ λ λ λ λ 2 1 2 22 21 1 12 11 2 1 2 22 21 1 12 11 What is the complexity?

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3. Matrix Multiplication = = = = = = = = = = n i ip mi i mi n i ip i n i i i n i i i n i ip i n i i i n i i i np n n p p mn m m n n b a b a b a b a b a b a b a b a B A b b b b b b b b b B a a a a a a a a a A 1 n 1 = i 1 1 2 1 2 2 1 1 2 1 1 1 2 1 1 1 1 2 1 2 22 21 1 12 11 2 1
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