Multiplying block matrices as 2x2 blocks

[el_hijoeputa]

Some big matrix, like 4x4 ones, can be written as groups of 2x2 matrices.

For example,
$$H = \left(\begin{array}{cccc}1 & 0 & 1 & 0\\0 & 1 & 0 & 1\\1 & 0 & 1 & 0\\0 & 1 & 0 & 1\end{array}\right) = \left(\begin{array}{cc}1 & 1\\1 & 1\end{array}\right)$$
where the 1 in the last matrix represents a 2x2 identity matrix.

I just want to know how to deal with this algebra the right way, and fast.
If A, B, C, D, W, X, Y, and Z are 2x2 matrices.

$$\left(\begin{array}{cc}A & B\\C & D\end{array}\right) \left(\begin{array}{cc}W & X\\Y & Z\end{array}\right) = \left(\begin{array}{cc}(AW+BY) & (AX+BZ)\\(CW+DY) & (CX+DZ)\end{array}\right)$$

Is this right?

 

[OlderDan]

Looks OK. You can see a general 4x4 times 4x4 done as 2x2 times 2x2 submatrices worked out here

http://www.cas.mcmaster.ca/~leduc/slides4f03/slides12.pdf

 

[thepatientmental]

Yes, this is correct. Grouping and multiplying 4x4 matrices can be simplified by breaking them down into 2x2 matrices and using the formula you have provided. This can make the calculation process faster and more efficient. However, it is important to make sure that the matrices are compatible for multiplication (i.e. the number of columns in the first matrix must equal the number of rows in the second matrix). Additionally, the order of multiplication matters, so it is important to pay attention to the order in which the matrices are multiplied. As long as these guidelines are followed, this method is a valid and efficient way to deal with larger matrices.