hw2 - I AB ) 2.3.b . Show that ( A I )( I B ) = ( I B )( A...

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Homework 2 Numerical Linear Algebra 1 Fall 2011 Solutions will be posted Wednesday, 9/21/11 Problem 2.1 Let x C n and y C n be two arbitrary vectors. Consider determining a circulant matrix C C n × n such that y = Cx 2.1.a . Assume that C exists for a given pair ( x, y ), show how to construct it. 2.1.b . When is C unique for a given pair ( x, y )? 2.1.c . When does C not exist for a given pair ( x, y )? Problem 2.2 Suppose x and y are two sparse vectors stored with their elements and indices in a compresed format that assumes the elements are stored in increasing order of their indices. Describe an algorithm to evaluate z x + y that does not make use of scatter/gather as in the notes. Compare the complexity of the two approaches. Problem 2.3 Let A, B, C, D C n × n be given square matrices. 2.3.a . Show that ( I A )( I B ) = (
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Unformatted text preview: I AB ) 2.3.b . Show that ( A I )( I B ) = ( I B )( A I ) = ( A B ) 2.3.c . Show that ( A I )( B I ) = ( AB I ) 2.3.d . Show that ( A B )( C D ) = ( AC ) ( BD ) . 2.3.e . Show that if A C n n and B C n n have inverses A-1 and B-1 , respectively, then ( A B ) has an inverse. 1 Problem 2.4 Let A C m m , B C n n , x C mn , and y C mn . 2.4.a . Describe an algorithm to evaluate the matrix vector product y = ( A B ) x i.e., given A, B, x determine y . 2.4.b . What is the complexity of the algorithm? 2.4.c . How does the complexity of the algorithm compare to the standard matrix-vector product computation? 2...
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This note was uploaded on 11/10/2011 for the course MAD 5932 taught by Professor Gallivan during the Fall '06 term at FSU.

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hw2 - I AB ) 2.3.b . Show that ( A I )( I B ) = ( I B )( A...

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