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math2J winter 2010 CLASS NOTES 3

math2J winter 2010 CLASS NOTES 3 - Class Notes from The...

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Class Notes from January 25, 2010 The homework for Thursday is Section 1.6 and Section 2.1. Today we ﬁnished Section 1.6 and started Section 2.1. We reviewed the three basic cases of matrix multiplication using block decompo- sitions of the matrices. Then we considered the fourth case of matrix multiplication using block decomposition. The goal is to simplify the computation of AB where A is an m × n matrix and B is an n × r matrix.The inverse can be expressed as the product of elementary matrices. Decompose A into four blocks A = ± A 11 A 12 A 21 A 22 ² (1) and similarly decompose B B = ± B 11 B 12 B 21 B 22 ² . (2) Now we want to multiply so you must decompose in such a way that A 11 and A 12 have s columns and B 11 and B 12 have s rows. Then A 21 and A 22 have n - s columns and B 21 and B 22 have n - s rows. This means the products A 11 B 11 , A 11 B 12 ,A 12 B 21 , A 11 B 12 , A 12 B 22 , A 21 B 11 , A 22 B 21 , A 21 B 12 and A 22 B 22 all make sense. The block multiplication formula in this case looks just like the formula for multiplying two 2

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math2J winter 2010 CLASS NOTES 3 - Class Notes from The...

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