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

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Homework 2 Numerical Linear Algebra 1 Fall 2010 Due date: beginning of class Monday, 9/27/10 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

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Unformatted text preview: . Show that ( I ⊗ A )( I ⊗ B ) = ( 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 . Describe an algorithm to evaluate the matrix vector product y = ( A ⊗ B ) x i.e., given A, B, x determine y . What is the complexity of your algorithm and how does it compare to the standard matrix-vector product computation? 2...
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