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# hw3 - Homework 3 Numerical Linear Algebra 1 Fall 2010 Due...

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Homework 3 Numerical Linear Algebra 1 Fall 2010 Due date: beginning of class Monday, 10/4/10 Problem 3.1 Problem 3.1.6 Golub and Van Loan p. 93 Problem 3.2 Recall that any unit lower triangular matrix L n × n can be written in factored form as L = M 1 M 2 · · · M n - 1 (1) where M i = I + l i e T i is an elementary unit lower triangular matrix (column form). Given the ordering of the elementary matrices, this factorization did not require any computation. Consider a simpler elementary unit lower triangular matrix (element form) that differs from the identity in one off-diagonal element in the strict lower triangular part, i.e., E ij = I + λ ij e i e T j where i = j . 3.2.a . Show that computing the product of two element form elementary matrices is simply superposition of the elements into the product given by E ij E rs = I + λ ij e i e T j + λ rs e r e T s whenever j = r . 3.2.b . Show that if j = r and i = s then computing E ij E rs with requires no compu- tation and E ij E rs = E rs E ij i.e., the matrices commute.

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hw3 - Homework 3 Numerical Linear Algebra 1 Fall 2010 Due...

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