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# 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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hw2 - I ⊗ AB 2.3.b Show that A ⊗ I I ⊗ B = I ⊗ B A...

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