How can I solve a matrix differential equation within MATLAB?

I am interested in solving an ODE dF/dt=F*A, where both A and F are matrices (in particular, 5x5 matrices). I have used ode45 and dsolve before for problems like dx/dt=A*x, where x is a vector but not a matrix like in this case.

EDIT:

I am using now a version of your example code with my 5x5 matrix A which is complex and defined as an anonymous function. Seems to work fine, is there any implemented method in MATLAB to check the solutions given by ode45? Thank you.

Answer 1

[James Tursa]

E.g., if you are using ode45, then simply reshape F and the initial Fo into column vectors. Inside the derivative routine, reshape the input argument F into a matrix, do your F*A matrix multiply, then reshape the multiply result back into a column vector for output.

EDIT:

Here is a small example showing the technique

function matrix_deriv_example
A = rand(2,2);  % Some arbitrary matrix we will use
F0 = eye(2);  % Some arbitrary matrix initial value
odefun = @(t,y) deriv(t,y,A);  % Anonymous derivative function with A
tspan = [0 5];
[T,F] = ode45(odefun,tspan,F0(:));  % Pass in column vector initial value
F = reshape(F.',2,2,[]);  % Reshape the output as a sequence of 2x2 matrices
n = size(F,3);
e = zeros(2,n);
for k=1:n
    e(:,k) = eig(F(:,:,k));  % Calculate the eigenvalues of the 2x2 matrices
end
plot(T,e(1,:),T,e(2,:)); grid on  % Plot them
xlabel('Time')
ylabel('Eigenvalues')
end

function dy = deriv(t,y,A)
F = reshape(y,size(A));  % Reshape input y into matrix
FA = F*A;  % Do the matrix multiply
dy = FA(:);  % Reshape output as a column vector
end

[Edu]

Ok, let's see if I understood well. Based on your answer, I would code something like this:

Reshape F and F0 into column vectors: F = F(:); F0 = F0(:);

Reshape F into matrix: F = reshape(F, size(A));

Why reshaping F in the previous step then??

F*A multiplication: dFdt = F*A;

Reshape result into column vector: dFdt = dFdt(:);

Use ode45 now?: [T F] = ode45(dFdt, [t0 tf], F0);

In case this is (close to be) correct, I have two questions:

1. How should I define F in the first place?
2. Can I use these method for a matrix A defined as an anonymous function?

Thanks very much.