# Simultaneously inverting many matrices

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Jeong Ho on 18 Jun 2015
Commented: Tohru Kikawada on 31 Jan 2021
Dear all, I have many 2-by-2 matrices (which are covariance matrices). I want to invert them all. I'm curious if there's an efficient way of doing this. I thought, maybe, you create a cell, in which each element is one of these matrices, and then use cellfun() in some way to do it. Quintessentially, my question is, is there a way of simultaneously inverting many matrices? I'd appreciate any and all comments. Thank you very much in advance!
Best, John

Walter Roberson on 18 Jun 2015
2 x 2 you might as well use the formula
D = A(1, 2, :) .* A(2, 1, :) - A(1, 1, :) .* A(2, 2, :);
V11 = -A(2, 2, :) ./ D;
V12 = A(1, 2, :) ./ D;
V21 = A(2, 1, :) ./ D;
V22 = -A(1, 1, :) ./ D;
invs = [V11, V12; V21, V22];

### More Answers (4)

Tohru Kikawada on 28 Apr 2019
Edited: Tohru Kikawada on 28 Apr 2019
You can leverage Symbolic Math Toolbox to vectorize the calculation.
% Define size of matrices
M=2;
N=10000;
A=rand(M,M,N);
% Calculate the inverse matrices in a loop
invA_loop = zeros(size(A));
tic
for k = 1:N
invA_loop(:,:,k) = inv(A(:,:,k));
end
disp('Elapsed time in calculation in a loop:');
toc
% Calculate the inverse matrices in a vectorization
As = sym('a', [M,M]); % Define an MxM matrix as a symbolic variable
invAs = reshape(inv(As),[],1); % Solve inverse matrix in symbol
invAfh = matlabFunction(invAs,'Vars',As); % Convert the symbolic function to an anonymous function.
tic
invA_sym = reshape(invAfh(A(1,1,:),A(2,1,:),A(1,2,:),A(2,2,:)),M,M,N);
disp('Elapsed time in the vectorized calculation:');
toc
% Max difference between the results in the loop and the vectorization
disp('Max difference between the elements of the results:');
disp(max(abs(invA_loop(:)-invA_sym(:))))
Results:
Elapsed time in calculation in a loop:
Elapsed time is 0.093938 seconds
Elapsed time in the vectorized calculation:
Elapsed time is 0.001396 seconds
Max difference between the elements of the results:
3.0323e-09
Alec Jacobson on 14 Nov 2020
%invA_sym = reshape(invAfh(A(1,1,:),A(2,1,:),A(1,2,:),A(2,2,:)),M,M,N);
Acell = reshape(num2cell(A,3),1,[]);
invA_sym = reshape(invAfh(Acell{:}),M,M,N);
so the code above works for M≠2
Tohru Kikawada on 31 Jan 2021
Alec, this is great! Thanks for your extension!

James Tursa on 18 Jun 2015
You might look at this FEX submission by Bruno Luong for solving 2x2 or 3x3 systems:

Hugo on 18 Jun 2015
In my experience, using cells is rather slow. Since your matrices are 2x2, then you could simple arrange them in a 3D array, with the first dimension representing the index of each matrix. Let's call this matrix M, which will be of size Nx2x2, N denoting the number of matrices you want to invert.
Now recall that the inverse of a matrix A=[A11,A12;A21,A22] can be computed as
[A22, -A12; -A21, A11] /(A11*A22-A12*A21)
You can implement this for all matrices as follows:
Minv = reshape([M(:,4),-M(:,2),-M(:,3),M(:,1)]./repmat(M(:,1).*M(:,4)-M(:,2).*M(:,3),1,4),N,2,2);
Hope this helps
Hugo

Azzi Abdelmalek on 18 Jun 2015
Using cellfun will not do it simultaneously. The for loop can be faster. But if you have a Parallel Computing Toolbox, you can do it with parfor

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