How can I perform multi-dimensional matrix multiplication in MATLAB?

For example, I have a 3-D matrix, "A", of size n1-by-n2-by-p, and a 2-D matrix, "B", of size n2-by-n1. For each page "p", I would like to calculate "B*A(:,:,p)", which should result in a new matrix of size n2-by-n2-by-p.

It is possible to perform this using a FOR loop, but FOR loops are not optimal for MATLAB calculations. Is there a more efficient way to do this?

Answer 1

[MathWorks Support Team]

For MATLAB versions before R2020b:

The ability to perform multi-dimensional matrix multiplication in MATLAB is not available.

As a workaround,  please use a FOR loop to achieve the intended multi-dimensional matrix multiplication.

For MATLAB versions R2020b and after:

You can use the "pagemtimes" MATLAB function, to perform page-wise (multi-dimensional) matrix multiplication. Please refer to the following documentation page for more information:

https://www.mathworks.com/help/matlab/ref/pagemtimes.html

For MATLAB versions R2022a and after:

In addition to "pagemtimes", and depending on your workflow, you can further use the "tensorprod" MATLAB function to perform tensor products. Please refer to the following documentation page for more information:

https://www.mathworks.com/help/matlab/ref/tensorprod.html

[Steven Lord]

I've alerted the MathWorks Support Team about this and asked them to update the post.

In addition to or instead of the pagemtimes function, depending on your application the tensorprod function introduced in release R2022a (which is listed in the See Also section of the pagemtimes documentation page) may also be of interest.

[Bruno Luong]

Also it would be good if the size is correct to n2 x n1 in the question example.

[Bruno Luong]

@MathWorks Support Team staff "which should result in a new matrix of size n1-by-n1-by-p."

Make it n2-by-n1-by-p.

[Haman]

@Bruno Luong Thank you for pointing out the error. We've updated the example from "B*A(:,:,p)*B" to "B*A(:,:,p)" and the size from "n1-by-n1-by-p" to "n2-by-n2-by-p" to correct the error in size.