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Unformatted text preview: row of A with the j th column of B ; that is, to Âµ a i 1 Â·Â·Â· a n 1 Â¶ b 1 j . . . b nj = a i 1 b 1 j + Â·Â·Â· + a n 1 b nj . 1 Note that, since all the numbers on the righthand side are nonnegative, the term on the righthand side is nonnegative. The sum of the entries in the i th row of AB is equal to n X j =1 ( a i 1 b 1 j + Â·Â·Â· + a n 1 b nj ) = a i 1 n X j =1 b 1 j + Â·Â·Â· + a n 1 n X j =1 b nj . Since B is a right stochastic matrix, we have n X j =1 b 1 j = Â·Â·Â· = n X j =1 b nj = 1 . Thus the sum of the entries in the i th row of AB is equal to a i 1 + Â·Â·Â· + a n 1 ; and since A is a right stochastic matrix, this is equal to 1. This the sum of the entries in each row of AB is equal to 1. 2...
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 Spring '09
 FokShuen
 Algebra, Matrices, Doubly stochastic matrix, Stochastic matrix, right stochastic matrix

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