Checking for singularity of matrix

General Usage
1

When doing linear algebra procedures such as
A\b or pinv(A),
where A is a square matrix, is there a way to check for singularity of A ? Otherwise I keep getting errors such as SingularException(3) or LAPACKException(1).

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2

rank

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3

pretty new to julia, but I think this would work:
is_singular = A → rank(A) == max(size(A)…)
#only true for matrix A when square and no dependent columns (full rank) = singular

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4

Numerically, this question is a bit ill-defined, because roundoff errors make it hard to distinguish between a matrix that is exactly singular and a matrix that is nearly singular. Moreover, as a practical matter it is rarely a good idea to try to make this distinction — matrices that are nearly singular are in many ways just as “bad” as matrices that are singular.

If you find yourself solving a lot of systems that are nearly singular, that it is a good sign that you need to re-think what you are doing. (e.g. perhaps you need some regularization in your equations)

Where are these matrices coming from? Why are they sometimes singular or nearly so?

5 Likes
5

Just adding that checking the condition number of a matrix can be done with:

julia> using LinearAlgebra

julia> M = rand(4,4)
4×4 Matrix{Float64}:
 0.596334  0.699729   0.632193   0.931758
 0.910922  0.150834   0.605005   0.90234
 0.640964  0.0686345  0.0811358  0.372114
 0.67283   0.458002   0.346649   0.485548

julia> cond(M)
23.45879906432043

julia> cond(zeros(4,4))     # zero matrix is singular
Inf

and for any singular matrix, the condition number would be Inf. A smaller condition number is better.

4 Likes
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Very nice point. I shall follow it.

7

Thank you, that is quite handy