math2J winter 2010 CLASS NOTES 2

math2J winter 2010 CLASS NOTES 2 - AB = ± A 1 B A 2 B ²(8...

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Class Notes from January 22, 2010 Today we covered Section 1.6. The topic was the computation of AB where A is an m × n and B is an n × r matrix using different blockdecompositions of A and B. A blockdecompositions of A is achieved by either drawing a horizontal line between rows of A or by drawing a vertical line between columns of A. Using the notation a 1 ,..., a m to denote the row vectors of A and b 1 ,.... , b r to denote the columns of B we can write A = a 1 a 2 . . . a m (1) or B = ( b 1 , ··· , b r ) . (2) Then matrix multiplication can be written as either AB = a 1 B a 2 B . . . a m B (3) or AB = ( A b 1 , ··· ,A b r ) . (4) We covered three block multiplications in class toady. A fourth one will be covered on Monday. CaseI Write A = ± A 1 A 2 ² (5) where A 1 = a 1 a 2 . . . a s (6) 1
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and A 2 = a s +1 a s +2 . . . a m (7) are s × n and ( m - s ) × n dimensional matrices so we can perform the multiplications A 1 B and A 2 B. Then the first block multiplication formula is
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Unformatted text preview: AB = ± A 1 B A 2 B ² . (8) CaseII The second formula comes from decomposing B = ( B 1 ,B 2 ) where B 1 = ( b 1 , ··· , b t ) (9) B 2 = ( b t + 1 , ··· , b r ) (10) are n × t and n × ( r-t ) matrices respectively. Thus we can form the products AB 1 and AB 2 and the second block multiplication formula is AB = ( AB 1 AB 2 ) . (11) CaseIII The last case comes from decomposing A = ± A 1 A 2 ² (12) where A 1 is m × s and A 2 is m × ( n-s ) and A = ( B 1 B 2 ) (13) with B 1 and s × r and B 2 an ( n-s ) × r matrix respectively. Notice we can form the matrix products A 1 B 1 and A 2 B 2 Then we have the third block multiplication formula AB = A 1 B 1 + A 2 B 2 . (14) 2...
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This note was uploaded on 01/27/2010 for the course MATH 2j taught by Professor Staff during the Winter '08 term at UC Irvine.

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math2J winter 2010 CLASS NOTES 2 - AB = ± A 1 B A 2 B ²(8...

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