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Multiplication Between Matrices Expressed in Row of Column Vectors or Column of Row Vectors University of British Columbia, Vancouver Yue-Xian Li February, 2014 1 A Matrix expressed as a row of column vectors An m × n matrix can be expressed as a row of n column vectors. Rules of matrix multiplication hold for such expressions. In the following text, no proof will be provided. Instead, examples will be provided to clarify the points. Example A1: A= 1 1 0 1 = [a1 a2 ] , where a1 = 1 0 , a2 = 1 1 . Thus, A 2 3 = [a1 a2 ] 2 3 = 2a1 + 3a2 = 2 1 0 +3 1 1 = 5 3 Let’s us verify the result using normal matrix multiplication: A 2 3 = 1 1 0 1 2 3 = 2 (1)(2) + (1)(3) (0)(2) + (1)(3) = 5 3 . . Summary: (1) ∀ m × n matrix A, one can express it as a row of column vectors aj ∈ Rm A = [a1 a2 · · · an ], (1 ≤ j ≤ n). (2) Let x = [x1 x2 · · · xn ]T be a column vector in Rn , then x1 x2 Ax = [a1 a2 · · · an ] .. = x1 a1 + x2 a2 + · · · + xn an . xn is a linear combination of column vectors of A. (3) Let B be a l × m matrix, then BA = B[a1 a2 · · · an ] = [Ba1 Ba2 · · · Ban ]. Since Baj (1 ≤ j ≤ n) is a linear combination of columns of B, each column of BA is a linear combination of columns of B. Example A2: If A is identical as in Example A1 and B = BA = [Ba1 Ba2 ] = [[b1 b2 ]a1 [b1 b2 ]a2 ] = [[b1 b2 ] 1 0 1 2 1 2 [b1 b2 ] = [b1 b2 ], then 1 1 3 ] = [b1 b1 + b2 ] = . 1 1 3 Notice that the first column of BA is equal to b1 while the second is equal to b1 + b2 . Conclusion: Columns of the matrix product Tn Tn−1 · · · T1 are linear combinations of columns of the leftmost (last) matrix Tn . 3 B Matrix expressed as a column of row vectors An m × n matrix can be expressed as a column of m row vectors. Rules of matrix multiplication hold for such expressions. Example B1: A= 1 1 0 1 = a1 a2 , a1 = [1 1] , a2 = [0 1] . Thus, [3 4] A = [3 4] a1 a2 = 3a1 + 4a2 = 3 [1 1] + 4 [0 1] = [3 7] . Let’s us verify the result using normal matrix multiplication: [3 4] A = [3 4] 1 1 0 1 = [(3)(1) + (4)(0) (3)(1) + (4)(1)] = [3 7] . Summary: (1) ∀ m × n matrix A, one can express it as a column of row vectors a1 a2 aTi ∈ Rn (1 ≤ i ≤ m). A = .. , . am (2) Let x = [x1 x2 · · · xm ] be a row vector in Rm , then a1 a2 xA = [x1 x2 · · · xm ] .. = x1 a1 + x2 a2 + · · · + xm am . am is a linear combination of row vectors of A. 4 (3) Let B be a n × l matrix, then AB = a1 a2 .. . B = am a1 B a2 B .. . . am B Since ai B (1 ≤ i ≤ m) is a linear combination of rows of B, each row of AB is a linear combination of rows of B. Example B2: If A is identical as in Example B1 and B = b 1 a1 b2 [1 1] a1 a1 B = AB = B= = b1 a2 a2 B a2 [0 1] b2 1 1 3 3 = b1 , then b2 b1 b2 = b1 + b2 = 4 4 . b1 b2 3 3 b2 Notice that the first row of AB is equal to b1 + b2 while the second is equal to b2 . Conclusion: Rows of the matrix product Tn Tn−1 · · · T1 are linear combinations of rows of the rightmost (first) matrix T1 . 5