rcond not correct for rank deficient matrix

Summary

I am using Eigen::FullPivLU with Eigen::Matrix<double, 7, 7>. I found that the rcond function of FullPivLU returns a non-zero value even if the input matrix is rank deficient. Maybe I have misunderstood something, but it seems to me that this is bug.

Best regards, Morten Åkerblad

Environment

• Operating System : Window
• Architecture : x64
• Eigen Version : 3.4.0
• Compiler Version : Visual Studio Version 17.11.6
• Compile Flags : -O3 -march=native
• Vector Extension : SSE/AVX/NEON ...

Minimal Example

using Matrix7x7 = Eigen::Matrix<double, 7, 7>;
using Vector7   = Eigen::Matrix<double, 7, 1>;

void test_rcond(Matrix7x7& A)
{
    Eigen::FullPivLU<Matrix7x7> lu_factorization(A);

    if (lu_factorization.rank() < 7)
    {
        double rcond = lu_factorization.rcond();
        assert(rcond == 0.0);
    }
}

Steps to reproduce

Here is a dump of the matrix A:

0.0000000000000000
0.0000000000000000
1.0000000000000000
0.0000000000000000
0.34149999916553497
0.0000000000000000
0.79877008515664061
0.0000000000000000
1.0000000000000000
0.0000000000000000
0.29200000315904617
0.0000000000000000
-0.37149999849498272
-0.16425902650844920
0.0000000000000000
0.0000000000000000
1.0000000000000000
0.0000000000000000
0.34149999916553497
0.0000000000000000
0.79877008515664061
0.0000000000000000
1.0000000000000000
0.0000000000000000
0.040500000119209290
0.0000000000000000
-0.30099999904632568
-0.081170580429391403
1.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
-0.0000000000000000
0.0000000000000000
0.70710689672598170
0.70710666564709435
0.027000000700354562
0.025455838867477515
-0.025455847186317101
-0.0068972271572272821
1.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
-0.0000000000000000

It has rank 5, but rcond returns 0.054420254040403307, which seems like a bug to me.

Here is another example where rank = 6 and rcond = 0.041005432281770185

	-0.18993746291019808
	0.0000000000000000
	0.0000000000000000
	1.0000000000000000
	0.0000000000000000
	0.35004772836926978
	0.0000000000000000
	0.78634840828157904
	0.0000000000000000
	1.0000000000000000
	0.0000000000000000
	0.0071360472582042700
	0.0000000000000000
	-0.30954772825006049
	-0.098431819579263849
	1.0000000000000000
	0.0000000000000000
	0.0000000000000000
	0.0000000000000000
	0.033363952861005020
	0.0000000000000000
	0.0031123239693799164
	0.0000000000000000
	1.0000000000000000
	0.0000000000000000
	-0.0063639521606504434
	0.0000000000000000
	-0.044547727654014047
	-0.011272635679894241
	0.70710689672598170
	0.0000000000000000
	-0.70710666564709435
	0.0000000000000000
	0.0000000000000000
	0.0000000000000000
	8.4516301250972170e-18

What is the current bug behavior?

What is the expected correct behavior?

Relevant logs

Warning Messages

Benchmark scripts and results

Anything else that might help

• Have a plan to fix this issue.