# 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
##### 2 CommentsShowHide 1 older comment
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)