A nonsingular matrix maybe very close to a singular one, e.g. if the absolute values of elements in a row are all very small, then such a matrix is nearly singular.
I can easily select a bunch of potential indices to quantitatively measure the nonsingularity of matrix, namely how far it is from being singular, e.g. the minimal modulus of eigenvalues or singular values, the absolute value of determinant...
My question is: Are there any documents which systematically discuss this issue, e.g. with comparisons between different indices?
Thanks a lot!
Hey,
Nice question. In telecommunications, there is a measure called conditional number
Conditional Number of a matrix is the absolute value of the ratio between the Maximum EigenValue to the Minimal EigenValue. As this ratio grow large (tends to infinity), this means that the matrix is tending towards a singular one( this is because the minimal eigenvalue is really small w.r.t the maximal eigenvalue); in that case the matrix is called ill-conditioned or not well conditioned. On the other hand, if this ratio goes towards 1, then the matrix is said to be well-conditioned.
General condition number is defined as norm(inv(A))*norm(A) where A is n/m matrix. For symmetric matrix this can be defined as max_eigenval/min_eigenval.
I know a book called Matrix Perturbation Theory which is mainly about the problem of nonsingularity of a matrix . If you are interested, please feel free to contact me.
I would like to slightly amend some of the previous posts: the condition number of a matrix A ( assumed square and real to keep things simple) is the ratio between the largest and the smallest singular values of A (recall the singular values of A are the eigenvalues of transpose(A)*A). When the condition number is very large, the matrix is nearly singular.
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