20105ee205A_1_2010ee205A_1_HW10_sol

20105ee205A_1_2010ee205A_1_HW10_sol - Chapter 13 Kronecker...

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Chapter 13 Kronecker Products 1. For any two matrices A and B for which the indicated matrix product is de±ned, show that (vec( A )) T (vec( B )) = Tr ( A T B ). In particular, if B IR n × n then Tr ( B ) = vec( I n ) T vec( B ). Answer 13.1 Suppose A is m × n . For the product A T B to be defned and square, B must be m × n as well. Now i± A and B are written in terms o± their columns, A =[ a 1 ,a 2 ,...,a n ] and B b 1 ,b 2 ,...,b n ] , then (vec( A )) T a T 1 a T 2 ...a T n ] and vec( B ) = [ b 1 2 n ] T . There- ±ore, (vec( A )) T (vec( B )) = a T 1 b 1 + a T 2 b 2 + · · · + a T n b n = Tr ( A T B ) . 2. Prove that for all matrices A and B ,( A B ) + = A + B + . Answer 13.2 Veri±y the ±our Penrose conditions: P1 ( A B )( A + B + )( A B ) = ( A B )( A + A B + B ) = ( AA + A ) ( BB + B )= A B . P2 ( A + B + )( A B )( A + B + ) = ( A + AA + ) ( B + + A + B + . P3 (( A B )( A + B + )) T = (( AA + ) ( + )) T =( AA + ) T ( + ) T = ( AA + ) ( + ) = ( A B )( A + B + ) . P4 (( A + B + )( A B )) T = (( A + A ) ( B + B )) T A + A ) T ( B + B ) T = ( A + A ) ( B + B ) = ( A + B + )( A B ) . There±ore, ( A B ) + = A + B + . 63
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64 CHAPTER 13. KRONECKER PRODUCTS 3. Show that the equation AXB = C is consistent for all C if A has full row rank and B has full column rank. Also, show that a solution, if it exists, is unique if A has full column rank and B has full row rank. What is the solution in this case? Answer 13.3 If a matrix M has full column rank, then it is one- one, and its pseudoinverse is M + =( M T M ) - 1 M T . If a matrix M has full row rank, then it is onto, and its pseudoinverse is M + = M T ( MM T ) - 1 . Thus, if A has full row rank and B has full column rank, then AA + CB + B = AA T ( AA
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This note was uploaded on 04/19/2011 for the course EE 205A taught by Professor Laub during the Fall '10 term at UCLA.

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20105ee205A_1_2010ee205A_1_HW10_sol - Chapter 13 Kronecker...

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