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Factorization into A = LU - MIT OpenCourseWare

Factorization into A = LU One goal of today s lecture is to understand Gaussian elimination in terms of matrices ; to find a matrix L such that A = LU. We start with some useful facts about matrix multiplication. Inverse of a product The inverse of a matrix product AB is B 1 A 1. Transpose of a product We obtain the transpose of a matrix by exchanging its rows and columns. In other words, the entry in row i column j of A is the entry in row j column i of AT . The transpose of a matrix product AB is BT AT . For any invertible matrix A, the inverse of AT is A 1 T.

elimination matrices Eij, so that A E21 A E31E21 A U. In the two by two case this looks like: → → →···→ E21 A U 1 0 2 1 2 1 −4 1 8 7 = 0 3 . We can convert this to a factorization A = LU by “canceling” the matrix E21; multiply by its inverse to get E−1 21 E21 A ...

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