Talk:Block matrix

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 To-do list for Block matrix: Add general formula for the determinant for ${\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}}$

I agree that the content under "Supermatrix" matches that under "partitioned matrices"

Merge

The actual merge discussion is over at Talk:Supermatrix. linas 20:50, 1 July 2006 (UTC)

Partitioning lines

I've always seen block matrices using partitioning lines (this article doesn't). So, for example, let A, B, C and D be n × n matrices; then the 2n × 2n block matrix is represented by

${\displaystyle \left({\begin{array}{c|c}A&B\\\hline C&D\end{array}}\right).}$ ~~ Dr Dec (Talk) ~~ 18:40, 22 November 2009 (UTC)
Mathworld doesn't. This Stanford page doesn't. This page doesn't. Hard to see that there is a universal convention; and the lines don't actually add anything. Charles Matthews (talk) 18:55, 22 November 2009 (UTC)
Charles, thank you so much for your warm, friendly and enlightening words. ~~ Dr Dec (Talk) ~~ 20:57, 24 November 2009 (UTC)

Block Toeplitz or Toeplitz Block matrices - order makes the difference!

It's a pity that Block Toeplitz matrices are called so, because this is a bit misleading. It would've made more sense to call Block Toeplitz matrices Toeplitz Block and call Block Toeplitz matices that have an arbitrary structure, but their blocks are Toeplitz matrices. (For instance, such are transition matrices for Markov chains, describing the extreme value of weight of gapped pairwise alignment of biological sequences) —Preceding unsigned comment added by 91.78.92.6 (talk) 01:27, 20 November 2010 (UTC)

Block Matrix Multiplication

For this formulation of block matrix multiplication to work, don't the cardinalities of the column partitions of A have to correspond to the cardinalities of the row partitions of B? Otherwise, the matrices in the A(alpha,gamma)B(gamma,beta) products will not be conformal. If this is correct, this section should be updated with this condition accordingly. (Fuug (talk) 02:30, 2 September 2012 (UTC))

You are correct. In the block matrix product of A and B, the partitioning of the matrices is not arbitrary but "the sizes of the submatrices of A and B [must be] such that the ... operations can be performed" (Howard Anton, Elementary Linear Algebra, page 36). Here's Anton's Problem 17(a), which shows this explicitly. Let
${\displaystyle A=\left({\begin{array}{ccc|c}-1&2&1&5\\0&-3&4&2\\\hline 1&5&6&1\end{array}}\right)\equiv \left({\begin{array}{c|c}A_{11}&A_{12}\\\hline A_{21}&A_{22}\end{array}}\right)}$
and
${\displaystyle B=\left({\begin{array}{cc|c}2&1&4\\-3&5&2\\\hline 7&-1&5\\0&3&-3\end{array}}\right)\equiv \left({\begin{array}{c|c}B_{11}&B_{12}\\\hline B_{21}&B_{22}\end{array}}\right)}$.
Ostensibly, the product ${\displaystyle AB}$ would be
${\displaystyle \left({\begin{array}{c|c}A_{11}B_{11}+A_{12}B_{21}&A_{11}B_{12}+A_{12}B_{22}\\\hline A_{21}B_{11}+A_{22}B_{21}&A_{21}B_{12}+A_{22}B_{22}\end{array}}\right)}$
but clearly many of those submatrix "products" like ${\displaystyle A_{11}B_{11}}$ have the wrong dimensions for matrix multiplication and cannot be performed. In fact, for this particular example, every intended product of submatrices has the wrong dimension. When I think of a non-awkward way to word this in the article, I will add it. Jason Quinn (talk) 20:10, 23 April 2013 (UTC)
I've updated the section in a way that I think captures precision while still being somewhat clear. Jason Quinn (talk) 20:31, 23 April 2013 (UTC)