MATLAB: Singular matrix and MATLAB inversion

inversesingular

How come that det(A) = 0, and yet MATLAB computes the inverse of A (i.e inv(A) is computed without any warnings).

Where A is a square symmetric matrix.

Thanks !

Best Answer

DON'T use det to determine if a matrix is singular!

A non-zero multiple of the identity matrix isn't singular, right?

A = 0.1*eye(400);
det(A)

The determinant of A underflowed to 0.

Can multiplying a singular (or nearly singular) matrix by a non-zero scalar make it no longer singular?

B = [1 1; 1 1+eps];
C = 1e8*B;
det(B)
det(C)

A number with a determinant on the order of 1 can't be singular, can it?

Use the cond or rcond functions instead. Matrices with condition numbers closer to 1 are better behaved.

cond(A)
cond(B)
cond(C)

If you're calling inv to try to solve a system of linear equations, DON'T. Use the backslash operator (\) instead. Even if your textbook told you to multiply by the inverse matrix, well, in theory there is no difference between theory and practice ...

Related Solutions

MATLAB: Chol() say matrix is not positive defnite even though all eigenvalues are positive

You're not factorizing the matrix you think you're factorizing. From the documentation for the chol function: "If A is nonsymmetric , then chol treats the matrix as symmetric and uses only the diagonal and upper triangle of A."

Is your Qd matrix symmetric?

issymmetric(Qd) % false

So what matrix are you actually trying to factorize?

A = triu(Qd) + triu(Qd, 1).'

That matrix is symmetric but is not positive definite.

MATLAB: How to tell if a random 3×3 Matrix is invertible

Back to your question, I have to produce a random 3x3 matrix A that is invertible and display it. One way could be to start with a matrix that you know will have a determinant of zero and then add random noise to each element. It worked for me to generate random matrices that are invertable.

for MC = 1:10000
    % first create a matrix that you know has a low rcond value:
    A = double(uint32(1000.*rand(3,1)).*uint32(1000.*rand(1,3)));
    % then add noise
    C = A + 100.0*rand(3,3);
    if (rcond(C)<1e-20)
        disp('algorithm fails');
        C
        inv(C)
    end
end

There were objections to this suggestion about checking the determinant value. I had said: If the determinant of a square matrix is 0, it can't be inverted. I'd suggestion to test with - using your tolerance on the last argument. See comments below.

Best Answer

Related Solutions

MATLAB: Chol() say matrix is not positive defnite even though all eigenvalues are positive

MATLAB: How to tell if a random 3×3 Matrix is invertible

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