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I am having a hard time understanding how to multiply blocked matrices with rectangle matrices and blocking into non-square matrices. Can someone please explain me how that works?

Zonta01
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  • Block matrix multiplication works just like regular matrix multiplication. And you can block a matrix however you want. –  Jun 04 '17 at 15:36
  • Can you explain more with a general formula? An example too would be very helpful please. – Zonta01 Jun 04 '17 at 15:37
  • Hi and welcome to Mathematics SE! A few tips/suggestions: Be more specific, and give some examples; let us know what you have tried yourself; and do some research on your own first. Perhaps this thread may help: https://math.stackexchange.com/questions/787909/block-matrix-multiplication . – Christopher.L Jun 04 '17 at 15:40

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I hope it can help you

Example:

$A= \begin{bmatrix} 2 & 3 & 2\\ 4 & 1 & 3\\ 3 & 2 & 1\\ \end{bmatrix} $$\to$ $ \left[ \begin{array}{cc|c} 2&3&2\\ 4&1&3 \\ \hline 3 &2 &1 \end{array} \right] $

$ B= \begin{bmatrix} 1 & 3 \\ 2 & 4 \\ 2 & 1 \\ \end{bmatrix} $$\to$ $ \left[ \begin{array}{} 1&3\\ 2&4 \\ \hline 2 &1 \end{array} \right] $

$AB= \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{bmatrix} \times\begin{bmatrix} B_{1} \\ B_{2} \\ \end{bmatrix} =\begin{bmatrix} A_{11}B_{1} + A_{12}B_{2} \\ A_{21}B_{1} + A_{22}B_{2} \\ \end{bmatrix} $

$A_{11}B_{1}= \begin{bmatrix} 2 & 3 \\ 4 & 1 \\ \end{bmatrix} \times\begin{bmatrix} 1 & 3 \\ 2 & 4 \\ \end{bmatrix} =\begin{bmatrix} 8 & 18 \\ 6 & 16 \\ \end{bmatrix} $

$A_{12}B_{2}= \begin{bmatrix} 2 \\ 3 \\ \end{bmatrix} \times\begin{bmatrix} 2 & 1 \\ \end{bmatrix} =\begin{bmatrix} 4 & 2 \\ 6 & 3 \\ \end{bmatrix} $

$A_{21}B_{1}= \begin{bmatrix} 3 & 2 \\ \end{bmatrix} \times\begin{bmatrix} 1 & 3 \\ 2 & 4 \\ \end{bmatrix} =\begin{bmatrix} 7 & 17 \\ \end{bmatrix} $

$A_{22}B_{2}= \begin{bmatrix} 1 \\ \end{bmatrix} \times\begin{bmatrix} 2 & 1 \\ \end{bmatrix} =\begin{bmatrix} 2 & 1 \\ \end{bmatrix} $

$A_{11}B_{1}+A_{12}B_{2}= \begin{bmatrix} 8 & 18 \\ 6 & 16 \\ \end{bmatrix} +\begin{bmatrix} 4 & 2 \\ 6 & 3 \\ \end{bmatrix} =\begin{bmatrix} 12 & 20 \\ 12 & 19 \\ \end{bmatrix} $

$A_{21}B_{1}+A_{22}B_{2}= \begin{bmatrix} 7 & 17 \\ \end{bmatrix} +\begin{bmatrix} 2 & 1 \\ \end{bmatrix} =\begin{bmatrix} 9 & 18 \\ \end{bmatrix} $

$$AB= \begin{bmatrix} 12 & 20 \\ 12 & 19 \\ \hline 9 & 18 \\ \end{bmatrix} $$

W.R.P.S
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