18-Pivoting-4UP

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 &amp; 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 &amp; columns of of A ( k = 1: n- 1)) are nonsingular....
View Full Document

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online