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M313Lec08 - Math 313 Lecture#8 1.6 Partitioned Matrices...

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Math 313 Lecture #8 § 1.6: Partitioned Matrices Matrix Multiplication Again. Recall that the ( i, j ) entry of the product AB for an m × n matrix A = ( a ij ) and a n × r matrix B = ( b ij ) is the scalar product (or inner product ) of the i th row of A and the j th column of B : n k =1 a ik b kj = a ( i, :) b j . Inherit in this is the partition of A into rows and B into columns. Recall also that each column of AB is the product of A with the corresponding column of B : AB = ( Ab 1 , . . . , Ab r ) . This results from another partition of A and B into blocks or submatrices : in this case, A is left unpartitioned (i.e., taken as itself), and B is partitioned into columns. Another partition of A and B gives a representation of AB in terms of its rows: AB = a (1 , :) B a (2 , :) B . . . a ( m, :) B . Example. Let A = 2 2 1 3 - 1 0 and B = 3 1 - 1 - 2 0 1 . Then the columns of AB are Ab 1 = 2 2 1 3 - 1 0 3 - 1 0 = 4 10 , Ab 2 = 2 2 1 3 - 1 0 1 - 2 1 = 1 5 , while the rows of AB are a (1 , :) B = 2 2 1 3 1 - 1 - 2 0 1 = 4 1 , a (2 , :) B = 3 - 1 0 3 1 - 1 - 2 0 1 = 10 5 . Although the partitions of A and B are different, each block multiplication leads to the same thing.
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Block Multiplication.
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