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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 defined, 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 defined and square, B must be m × n as well. Now if A and B are written in terms of 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 , b 2 , . . . , b n ] T . There- fore, (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 Verify the four 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 + BB + ) = A + B + . P3 (( A B )( A + B + )) T = (( AA + ) ( BB + )) T = ( AA + ) T ( BB + ) T = ( AA + ) ( BB + ) = ( 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 ) . Therefore, ( 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
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20105ee205A_1_2010ee205A_1_HW10_sol - Chapter 13 Kronecker...

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