math2J winter 2010 CLASS NOTES 2

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online