18-Pivoting-4UP

18-Pivoting-4UP - Permutation matrices Pivoting Solving...

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Unformatted text preview: Permutation matrices Pivoting Solving systems Pivoting Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Pivoting MATH 2070U 1 / 23 Permutation matrices Pivoting Solving systems Pivoting 1 Permutation matrices and LU factorisation 2 Partial pivoting 3 Solving systems of equations using decomposition PA = LU c D. Aruliah (UOIT) Pivoting MATH 2070U 2 / 23 Permutation matrices Pivoting Solving systems Permutation matrices Definition (Permutation matrix) A permutation matrix is any matrix obtained from interchanging the rows or columns of the identity matrix. e.g., P = 1 1 P = 1 1 1 , P = 1 1 1 1 For any permutation matrix, P- 1 = P T PA permutes rows of matrix A AP T permutes columns of matrix A Permutation matrices can be stored as single vector c D. Aruliah (UOIT) Pivoting MATH 2070U 4 / 23 Permutation matrices Pivoting Solving systems Need for pivoting Not every invertible matrix A has LU factorisation A = LU Example (Pivoting mandatory) A = 1 1 1 A 1,1 = prevents direct elimination Instead, use permutation matrix to swap rows 1 & 2 PA = 1 1 1 if P = 1 1 PA is upper triangular, so LU factorisation is PA = LU with P = 1 1 , L = 1 1 and U = 1 1 1 c D. Aruliah (UOIT) Pivoting MATH 2070U 5 / 23 Permutation matrices Pivoting Solving systems Proposition For a given nonsingular matrix A R n n , the Gauss factorisation A = LU exists and is unique iff all the principal submatrices A k of A of order k (that is those obtained by restricting the first k rows & columns of of A ( k = 1: n- 1)) are nonsingular....
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18-Pivoting-4UP - Permutation matrices Pivoting Solving...

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