hw1_2011_04_06_01_solutions

hw1_2011_04_06_01_solutions - EE263 S Lall 2011.04.06.01...

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EE263 S. Lall 2011.04.06.01 Homework 1 Solutions Due Thursday 4/7. 1. Interpretations of matrix multiplication. 0101 Let B be a 4 × 4 matrix, to which we apply the following operations: (i) double column 1, (ii) halve row 3, (iii) add row 3 to row 1, (iv) interchange columns 1 and 4, (v) subtract row 2 from each of the other rows, (vi) replace column 4 by column 3, (vii) delete column 1 (so that the column dimension is reduced by 1) (a) Write the result as a product of 8 matrices. (b) Write it again as a product ABC (same B ) of three matrices. Solution. (a) Each of the operations can be performed as follows. (a) Double column 1: let B be B = bracketleftbig b 1 b 2 b 3 b 4 bracketrightbig where b i is a column vector in R 4 . Then postmultiply by a diagonal matrix as follows. H = bracketleftbig b 1 b 2 b 3 b 4 bracketrightbig 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright W = bracketleftbig 2 b 1 b 2 b 3 b 4 bracketrightbig (b) Halve row 3: let the rows of H be H = h T 1 h T 2 h T 3 h T 4 then J = 1 0 0 0 0 1 0 0 0 0 1 2 0 0 0 0 1 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright P h T 1 h T 2 h T 3 h T 4 = h T 1 h T 2 1 2 h T 3 h T 4 (c) Add row 3 to row 1: premultiply J as follows. K = 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright Q j T 1 j T 2 j T 3 j T 4 = j T 1 + j T 3 j T 2 j T 3 j T 4 1
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EE263 S. Lall 2011.04.06.01 (d) Interchange column 1 and 4 L = bracketleftbig k 1 k 2 k 3 k 4 bracketrightbig 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright R = bracketleftbig k 4 k 2 k 3 k 1 bracketrightbig (e) Subtract row 2 from each of the other rows M = 1 1 0 0 0 1 0 0 0 1 1 0 0 1 0 1 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright S l T 1 l T 2 l T 3 l T 4 = l T 1 l T 2 l T 2 l T 3 l T 2 l T 4 l T 2 (f) Replace column 4 by column 3 N = bracketleftbig m 1 m 2 m 3 m 4 bracketrightbig 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright T = bracketleftbig m 1 m 2 m 3 m 3 bracketrightbig (g) Delete column 1 (so that the column dimension is reduced by 1) A = bracketleftbig n 1 n 2 n 3 n 4 bracketrightbig 0 0 0 1 0 0 0 1 0 0 0 1 bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright U = bracketleftbig n 2 n 3 n 4 bracketrightbig Then we have the result as the following product of 8 matrices A = SQPBWRTU (b) Write it again as a product ABC (same B ) of three matrices.
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