CP07_matrix - Matrices An mn matrix is a rectangular array of complex or real numbers arranged in m rows and n columns Types Operations etc Types

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1 Matrices Matrices An m×n matrix is a rectangular array of complex or real numbers arranged in m rows and n columns: mn m m m n n n a a a a a a a a a a a a a a a a K K K K K K K K K 3 2 1 3 33 32 31 2 23 22 21 1 13 12 11 2 Types, Operations, etc. Types: square, symmetric, diagonal, Hermithean, … Basic operations: A+B, A-B, AB (AB BA). Square matrices Determinant: det(A) Inverse matrix A -1 : AA -1 = I (I is a unit matrix) 3 Applications Linear systems of equations Eigenvalue problem 4 Linear systems of equations m>n over determined system (data processing) m=n square case (what we will do) m<n under determined system 5 Linear systems in matrix notation or Ax = b 6 Two cases for right-hand coefficients Ö right-hand coefficients b i 0 Unique solution if the determinant det(A) 0 Ö right-hand coefficients b i =0 Unique solution if the determinant det(A) = 0
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7 Analytic solutions for n=2 a 11 x 1 + a 12 x 2 =b 1 a 21 x 1 + a 22 x 2 =b 2 expressing the first unknown x 1 from the first equation x 1 = (b 1 -a 12 x 2 )/a 11 and substituting to the second equation we have a single equation with one unknown x 2 . 8 Gaussian elimination Ö Since there is no such an operator as elimination neither in C++ nor Fortran we should translate this procedure to an appropriate numerical method for solving systems of linear equations. Ö Numerical method = Gaussian elimination 9 Gaussian elimination for n=3 Let subtract the first equation multiplied by the coefficient a 21 /a 11 from the second one, and multiplied by the coefficient a 31 /a 11 from the third equation. 10 Step 2: Step 2: Repeating the same procedure to the last of two equations gives where 11 Step 3: Step 3: Doing back substitution we will find x 2 and then x 1 . This direct method to find solutions for a system of
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This note was uploaded on 09/29/2009 for the course PHYSICS 811 taught by Professor Godunov during the Fall '09 term at Old Dominion.

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CP07_matrix - Matrices An mn matrix is a rectangular array of complex or real numbers arranged in m rows and n columns Types Operations etc Types

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