Multiplying matrix polynomials in MATLAB

Question

I'm trying to perform matrix polynomial multiplication of the type:

(A_1+A_2*y)*(B_1+B_2*y+B_3*y^2)

where y is the variable, all the A_i are constant matrices of the same size and all the B_i are constant matrices of the same size, and the matrix multiplication of the form A_i*B_i makes sense. The matrices A_i and B_i are known and specified by the user.

Formally the multiplication should result in C_1+C_2*y+C_3*y^2+C_4*y^3 and I would like to know the C_i. In MATLAB, if the A_i and B_i are scalars, one could use 'conv'. However, if they are not scalars, the problem is not as easy. I would like to know if there is a simple way (analogous to 'conv') of doing this with matrices in MATLAB: given the A_i and B_i I would like to know the C_i? Of course, I'd like this in a general sense (univariate matrix polynomials of any degree), and would highly want to avoid using the symbolic toolbox.

I though I understood your question, but in fact is not quite clear. What do you want to do with the matrices A_1, A_2 and so on? Do you want them to be scalars? Subreferences from a matrix? What have you tried? — Werner
@Werner Yes, an example would have been illuminating. What he wants is an equivalent to the scalar conv, an example: a = [1,1]; % <-> 1+1x, b = [1,1]; % <-> 1+1x, c = conv(a,b); % = [1,2,1] <-> (1+1x)*(1+1x) = 1+2x+1x^2. Take a look at the end of my "Test by" code to understand what he is talking about :-) — matheburg

matheburg matheburg · Accepted Answer · 2013-08-14T19:47:27

It's probably not the best solution, but you could reimplement conv for matrices, e.g.:

function C = convMat(A,B)
    nA = size(A,3);
    nB = size(B,3);
    n = nA + nB - 1;
    C = zeros([size(A,1),size(B,2),n]);
    for k = 1:n
        for j = max(1,k+1-nB):min(k,nA)
            C(:,:,k) = C(:,:,k) + A(:,:,j)*B(:,:,k-j+1);
        end
    end

For your interest: Another (worse?) implementation:

function C = convMat2(A,B)
    n = size(A,3) + size(B,3) - 1;
    C = zeros([size(A,1),size(B,2),n]);
    for mA = 1:size(A,1)
        for mB = 1:size(B,2)
            for l = 1:size(A,2) % = size(B,1)
                vA = A(mA,l,:);
                vA = vA(:);
                vB = B(l,mB,:);
                vB = vB(:);
                C(mA,mB,:) = C(mA,mB,:) + reshape(conv(vA,vB),[1,1,n]);
            end
        end
    end

Test by:

% matrix example
A(:,:,1) = rand(3,3); % A1
A(:,:,2) = rand(3,3); % A2
B(:,:,1) = rand(3,4); % B1
B(:,:,2) = rand(3,4); % B2
B(:,:,3) = rand(3,4); % B3

C1 = convMat(A,B);
C2 = convMat2(A,B);

% test
x = rand(1,1);
(A(:,:,1) + A(:,:,2)*x) * (B(:,:,1) + B(:,:,2)*x + B(:,:,3)*x^2)
C1(:,:,1) + C1(:,:,2)*x + C1(:,:,3)*x^2 + C1(:,:,4)*x^3
C2(:,:,1) + C2(:,:,2)*x + C2(:,:,3)*x^2 + C2(:,:,4)*x^3

I am sure it is possible to optimize convMat.

Multiplying matrix polynomials in MATLAB

2 Answers