NumericalMethodsChaper01

NumericalMethodsChaper01 - 1 1. Numerical Methods - Chapter...

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Unformatted text preview: 1 1. Numerical Methods - Chapter 1 1.1. Review of Matrix Algebra A matrix is rectangular array of numbers. = m n m n n n n m n m n n n n m m m m m m a a a a a a a a a a a a a a a a a a a a a a a a a A 1 , 3 2 1 , 1 1 , 1 3 , 1 2 , 1 1 , 1 3 1 , 3 33 32 31 2 1 , 2 23 22 21 1 1 , 1 13 12 11 ] [ L L M M O M M M L L L [ A ] is an n m matrix, where: n = number of rows m = number of columns The element of the matrix at row i and column j is: a ij where: i = 1, 2, 3, , n j = 1, 2, 3, , m i , j are called indicies Square matrix: n = m Column matrix = Column vector = vector: m = 1 a i 1 = a i Row matrix = row vector = transpose of a vector = n = 1 a 1 j = a j A scalar can be thought of as a matrix with one row and one column, i.e, n = m = 1 1.1.1. Matrix addition Matrix notation: ] [ ] [ ] [ B A C + = + + + + + + + + + + + + + + + + + + + + + + + + + = m n m n m n m n n n n n n n m n m n m n m n n n n n n n m m m m m m m m m m m m m n m n n n n m n m n n n n m m m m m m b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a c c c c c c c c c c c c c c c c c c c c c c c c c 1 , 1 , 3 3 2 2 1 1 , 1 , 1 1 , 1 1 , 1 3 , 1 3 , 1 2 , 1 2 , 1 1 , 1 1 , 1 3 3 1 , 3 1 , 3 33 33 32 32 31 31 2 2 1 , 2 1 , 2 23 23 22 22 21 21 1 1 1 , 1 1 , 1 13 13 12 12 11 11 1 , 3 2 1 , 1 1 , 1 3 , 1 2 , 1 1 , 1 3 1 , 3 33 32 31 2 1 , 2 23 22 21 1 1 , 1 13 12 11 L L M M O M M M L L L L L M M O M M M L L L Matrix [ A ] and matrix [ B ] must have the same number of rows and columns, i.e. both matrices must be of size n m Indicial notation: ij ij ij b a c + = Subtraction: ] [ ] [ ] [ B A C = ij ij ij b a c = Properties of matrix addition: ] [ ] [ ] [ ] [ ] [ A B B A C + = + = ij ij ij ij ij a b b a c + = + = 1.1.2. Multiplication by a scalar ] [ ] [ ] [ A A C = = ij ij a c = 1.1.3. Transpose of a matrix T A C ] [ ] [ = ji ij a c = 2 = mn n m n n n n m n m n n n m m m m m m m n m n n n n m n m n n n n m m m m m m a a a a a a a a a a a a a a a a a a a a a a a a a c c c c c c c c c c c c c c c c c c c c c c c c c , 1 3 2 1 1 , 1 , 1 1 , 3 1 , 2 1 , 1 3 3 , 1 33 23 13 2 2 , 1 32 22 12 1 1 , 1 31 21 11 1 , 3 2 1 , 1 1 , 1 3 , 1 2 , 1 1 , 1 3 1 , 3 33 32 31 2 1 , 2 23 22 21 1 1 , 1 13 12 11 L L M M O M M M L L L L L M M O M M M L L L If [ A ] is an n m matrix, then [ C ] is an m n matrix. matrix....
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NumericalMethodsChaper01 - 1 1. Numerical Methods - Chapter...

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