The linear transformation X À Ä given by XÐBÑ œ EB is invertible À that is, there

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1 What do we know about an invertible ( 8 8) matrix E? E is invertible Í E rref ÐEÑ œ M 8 X is invertible X À Ä (where XÐBÑ œ EB) is onto for every, in 8, EB œ, has at least " solution The columns of E span 8 E has a pivot position in every row (****) ( 8 8) E has a pivot position in every column (****) rref ÐEÑ œ M 8 for every, in 8, EB œ, has at most " solution EB œ! has only trivial solution X À Ä (where XÐBÑ œ EB) is one-to-one E has linearly independent columns The (equivalent) blue statements are all related to a pivot position in every row. The (equivalnet) red statements are all related to a pivot position in every column All of the statements are equivalent for a square matrix E because, then, a pivot in every row and a pivot in every column are equivalent A few other statements equivalent to all of the above (disscused in class/textbook) There is a 88 8 matrix H such that EH œ M There is a 88 8 matrix G such that GH œ M The linear transformation X À Ä given by XÐBÑ œ EB is invertible À that is, there is a linear transformation W À Ä such thatðw XÑÐBÑ œ B and ÐX WÑÐBÑ œ B for all B in 8 Þ The formula isx " ÐBÑ œ WÐBÑ œ E " BÞ

2 Suppose E, F are the correct sizes to compute EFÞ If we partition E and F into blocks so that column partitions of E match sizes with the row partitions in Fß then can compute product EF by using the row-column rule on the blocks as if they were scalars. In this example, the column partition of E is three-one. The same is true for the row partition of B. * * * * * * * * Ö * * * * Ù * * * * ÙÖ Ö ÙÖ * * * * Ö Ù Õ Ø _ * * * * Õ * * * * Ø $ % % % Æ Æ E"" E"# F"" F"# E F #" E## #" F## # # block matrix # 2 block matrix # # # # E"" F"" E "# F #" E"" F"# E"# F## œ E F E F E F E F #" "" ## #" #" "# ## ## â â " # " # Answer is a # # block matrix, or $ % without the partitioning

3 Example (here the column partition sizes in E are three-one-two, and the rows are partitioned similiarly in a three-one-two pattern.) "! " l " # $! l! " # # l " l 1 1 Ö "! " l " Ù! "! l! l! 1 ÙÖ ÙÖ ÙÖ Ö ÙÖ! #! l #! " # l " l #! Õ Ø!!! l " l! " Ö Ù " "! l " Õ!! # l # Ø E"" F"" E"# F#" E"$ F$" ) * & 8 2 = # $ # Ö Ù (# # block matirx) ' ( # ' Õ! # # % Ø or Ö Ù (% % without partitioning) Õ Ø

4 Example E F H Æ Æ Æ " # * * * * ( ) * "! $ % Õ& ' Ø "" "# "$ "% œ * * * * Õ * * * * Ø Here the column partition sizes for E are one-one, and the row particion sizes for F are one-one. " G" œ $ ( ) * "!Óß Õ& Ø V " œ Ò # G# œ % V# œ "" "# "$ "%Ó Õ' Ø ß Ò V" The blocks are ÒG" G# Ó and and these are shaped so that all the products below V # make sense: H œ G " V " G V Å Å $ % $ % # # ( ) * "! #& #% #' #) œ #" #% #( $! %% %) &# &' Õ$& %! %& &! Ø Õ'' (# () )% Ø #* $# $& $) œ '& (# (* )' Õ"!" ""# "#$ "$% Ø

5 Same example, general version 8 columns Suppose E is 7 8 and F is 8 : ß 8 rows Let G",..., G8 be the columns of E (each 7 " ) V ßÞÞÞßV be the rows of F (each " :) " 8 then EF œ G V ÞÞÞÞ G V " " â ß all 7 : (**) (**) is the Column-Row Expansion of EF Partitioning matrices into smaller blocks for multiplication can sometimes give some theoretical insights. But in addition, if it were necessary to multiply EF where, say, each of E and F had several millino rows and columns, the the job might be too large for your computer to handle in one bite. By partitioning E and F into smaller blocks, the pieces to be multiplied may be of more manageable size. (In addition, in applications, the matrices E and F might be sparse meaning that a large fraction of the entries are!. In such a case, the partitioning may create many blocks of!'s and muiltiplications that involve those blocks are triival. (See the introductory application page at the start of Chapter 2 about computational fluid dynamics and mathematical modelling in the design of an aircraft body.