# How to multiply higher order matrices?

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Nathan Zechar on 24 May 2021
Answered: Nathan Zechar on 25 May 2021
Hello, I am attempting to multiply a 5D 3x3x3x3x3 matrix and a 5D 3x3x3x3x1 matrix in order to produce a 4D matrix of 3x3x3x3 after squeeze operation is applied.
Here's is what it would look like in an equation.
Multiplying a 3x3 by a 3x1 works no problem,
A = ones(3,1);
B = ones(3,3);
C = B*A;
However when I try to extend this to higher orders, I recieve and error
A = ones(3,3,3,3,1);
B = ones(3,3,3,3,3);
C = B*A;
The error suggests I use pagemtimes, but I'm not so sure those are the correct operations for my application. How would I go about coding this correctly?
EDIT: I believe I might have found the right answer, but not sure. The code below leads me to believe this might be correct
A = ones(3,1,3);
B = ones(3,3,3);
C = pagemtimes(B,A);
which would then exent to
A = ones(3,1,3,3,3);
B = ones(3,3,3,3,3);
C = pagemtimes(B,A);
EDIT2: Yes the above works, you can check it will the code below
clear all
a = 10;
%% Convert 3D matrix to 5D for tensor matrix multiplication
Bx = 1*ones(a,a,a);
By = 2*ones(a,a,a);
Bz = 3*ones(a,a,a);
B(1,1,:,:,:) = Bx;
B(2,1,:,:,:) = By;
B(3,1,:,:,:) = Bz;
%% Generate 5D tensor matrix
exx = 1*ones(a,a,a);
exy = 2*ones(a,a,a);
exz = 3*ones(a,a,a);
eyx = 4*ones(a,a,a);
eyy = 5*ones(a,a,a);
eyz = 6*ones(a,a,a);
ezx = 7*ones(a,a,a);
ezy = 8*ones(a,a,a);
ezz = 9*ones(a,a,a);
eT(1,1,:,:,:) = exx;
eT(1,2,:,:,:) = exy;
eT(1,3,:,:,:) = exz;
eT(2,1,:,:,:) = eyx;
eT(2,2,:,:,:) = eyy;
eT(2,3,:,:,:) = eyz;
eT(3,1,:,:,:) = ezx;
eT(3,2,:,:,:) = ezy;
eT(3,3,:,:,:) = ezz;
%% Multiply 3x3 3D matrix by 3x1 3D matrix
A = pagemtimes(eT,B);
%% Convert 5D matrix into 3 3D matrix components
Ax = squeeze(A(1,1,:,:,:));
Ay = squeeze(A(2,1,:,:,:));
Az = squeeze(A(3,1,:,:,:));
%% Ax should = 14 at each location
%% Ay should = 32 at each location
%% Az should = 50 at each location
Torsten on 25 May 2021
Is there any standard definition for the multiplication of matrices of dimension greater than 2 ?
I think no. So you will have to program the operation on your own.
Nathan Zechar on 25 May 2021
I'm not sure, but my orginal 2nd edit was in fact the solution after I checked it.
I posted the code. Thanks!

Nathan Zechar on 25 May 2021
Here's an example. This is one way of doing it.
clear all
a = 10;
%% Convert 3D matrix to 5D for tensor matrix multiplication
Bx = 1*ones(a,a,a);
By = 2*ones(a,a,a);
Bz = 3*ones(a,a,a);
B(1,1,:,:,:) = Bx;
B(2,1,:,:,:) = By;
B(3,1,:,:,:) = Bz;
%% Generate 5D tensor matrix
exx = 1*ones(a,a,a);
exy = 2*ones(a,a,a);
exz = 3*ones(a,a,a);
eyx = 4*ones(a,a,a);
eyy = 5*ones(a,a,a);
eyz = 6*ones(a,a,a);
ezx = 7*ones(a,a,a);
ezy = 8*ones(a,a,a);
ezz = 9*ones(a,a,a);
eT(1,1,:,:,:) = exx;
eT(1,2,:,:,:) = exy;
eT(1,3,:,:,:) = exz;
eT(2,1,:,:,:) = eyx;
eT(2,2,:,:,:) = eyy;
eT(2,3,:,:,:) = eyz;
eT(3,1,:,:,:) = ezx;
eT(3,2,:,:,:) = ezy;
eT(3,3,:,:,:) = ezz;
%% Multiply 3x3 3D matrix by 3x1 3D matrix
A = pagemtimes(eT,B);
%% Convert 5D matrix into 3 3D matrix components
Ax = squeeze(A(1,1,:,:,:));
Ay = squeeze(A(2,1,:,:,:));
Az = squeeze(A(3,1,:,:,:));
%% Ax should = 14 at each location
%% Ay should = 32 at each location
%% Az should = 50 at each location

James Tursa on 25 May 2021
You may need to permute your arrays first to get your desired 2D slices in the first two dimensions.
A = whatever
B = whatever
Ap = permute(A,[4 5 1 2 3]);
Bp = permute(B,[4 5 1 2 3]);
Then use pagemtimes
C = pagemtimes(Bp,Ap);
Then if needed you can permute the answer back
C = permute(C,[3 4 5 1 2]);

Sulaymon Eshkabilov on 25 May 2021
Hi,
Maybe this one solves your exercise: squeeze(C)