MATLAB: What should be used to diagonalise a complex sparse matrix instead of ‘eig’

eigMATLAB

I want to diagonalise a sparse and complex matrix. In the documentation page of 'eig', it should be used for sparse and real matrices. What else should be used for a sparse and complex matrix? 'eigs'?

I tried one thing: I converted the sparse matrix into a full matrix and used 'eig' for it. However, its eigenvalues showed slightly wrong values. Any idea to fix this?

Best Answer

If converting the sparse matrix into a dense matrix is possible, that seems like the best thing to do. How do you measure that the eigenvalues are slightly wrong?

Related Solutions

MATLAB: Diagonalising a Skew-Symmetric Matrix

When EIG is called with an exactly symmetric/hermitian matrix, MATLAB falls back to a specialized algorithm that guarantees that U is orthogonal/unitary and that D is real. There is no such special algorithm choice for skew-symmetric matrices, so there is no guarantee here, even though if the problem is nicely conditioned, the result will be close to that:

>> rng default; A = randn(10); A = A - A';
>> [U, D] = eig(A);
>> max(abs(real(diag(D))))
ans =
   2.1034e-16
>> norm(U'*U - eye(10))
ans =
   4.7239e-15

However, if matrix B is (exactly) skew-symmetric, it implies that matrix A = 1i*B is hermitian, and passing this matrix to EIG will result in unitary eigenvectors and all-real eigenvalues, which you can then transform back:

[U, D] = eig(1i*A);
D = -1i*D;
>> max(abs(real(diag(D))))
ans =
     0
>> norm(U'*U - eye(10))
ans =
   1.5001e-15

Best Answer

Related Solutions

MATLAB: Diagonalising a Skew-Symmetric Matrix

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