# Error in NullSpace, with AlgebraicNumber entries

Consider the following 14x14 matrix, with typical entry

AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{1/13,0,0,-(2/13),0,0,0,0,0,0,2/13,0,0,0,0,0,-(2/13),0,0,0,0,0,0,0,0,2/13,0,0,0,-(2/13),2/13,0,0,0,0,2/13,0,0,0,0,2/13,0,-(2/13),0,0,0,0,0}]


Mathematica (9.0.1.0, on OS X) claims that the matrix rank is 12:

In[2] := MatrixRank[M]
Out[2] := 12


and indeed happily produces the null space, claiming for example that the following vector is in the nullspace:

In[3] := n = NullSpace[M][[1]]
Out[3] := {AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{5/3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,4/3,0,0,0,0,2/3,0,0,0,0,2/3,0,0,0,0,4/3,0,0,0,0,2,0,0}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{8/3,0,0,8/3,0,4/3,0,0,-(4/3),0,-(8/3),0,0,0,0,0,8/3,0,0,0,8/3,-(4/3),0,0,0,2,0,0,0,8/3,-2,0,0,0,-(4/3),-2,0,0,0,0,2,0,8/3,0,0,8/3,0,-(4/3)}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{0,0,0,-(2/3),0,-(2/3),0,0,2/3,0,2/3,0,0,0,0,0,-(2/3),0,0,0,2/3,2/3,0,0,0,0,0,0,0,-(2/3),2/3,0,0,0,2/3,2/3,0,0,0,0,0,0,-(2/3),0,0,2/3,0,2/3}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{8,0,0,14/3,0,8/3,0,0,-(8/3),0,-(14/3),0,0,0,0,0,14/3,0,0,0,22/3,-(8/3),0,0,0,6,0,0,0,14/3,-(8/3),0,0,0,-(8/3),-(8/3),0,0,0,0,6,0,14/3,0,0,22/3,0,-(8/3)}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{12,0,0,10/3,0,4,0,0,-4,0,-(10/3),0,0,0,0,0,10/3,0,0,0,10,-4,0,0,0,20/3,0,0,0,10/3,-(10/3),0,0,0,-4,-(10/3),0,0,0,0,20/3,0,10/3,0,0,10,0,-4}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{8/3,0,0,0,0,2/3,0,0,-(2/3),0,0,0,0,0,0,0,0,0,0,0,10/3,-(2/3),0,0,0,2,0,0,0,0,2/3,0,0,0,-(2/3),2/3,0,0,0,0,2,0,0,0,0,10/3,0,-(2/3)}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{6,0,0,10/3,0,2,0,0,-2,0,-(10/3),0,0,0,0,0,10/3,0,0,0,6,-2,0,0,0,14/3,0,0,0,10/3,-(4/3),0,0,0,-2,-(4/3),0,0,0,0,14/3,0,10/3,0,0,6,0,-2}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{-(20/3),0,0,-(8/3),0,-2,0,0,2,0,8/3,0,0,0,0,0,-(8/3),0,0,0,-6,2,0,0,0,-(14/3),0,0,0,-(8/3),2,0,0,0,2,2,0,0,0,0,-(14/3),0,-(8/3),0,0,-6,0,2}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{-(20/3),0,0,-4,0,-(8/3),0,0,8/3,0,4,0,0,0,0,0,-4,0,0,0,-(22/3),8/3,0,0,0,-6,0,0,0,-4,4/3,0,0,0,8/3,4/3,0,0,0,0,-6,0,-4,0,0,-(22/3),0,8/3}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{-(2/3),0,0,-(4/3),0,-(2/3),0,0,2/3,0,4/3,0,0,0,0,0,-(4/3),0,0,0,-(4/3),2/3,0,0,0,-(2/3),0,0,0,-(4/3),2/3,0,0,0,2/3,2/3,0,0,0,0,-(2/3),0,-(4/3),0,0,-(4/3),0,2/3}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{-(14/3),0,0,-(8/3),0,-2,0,0,2,0,8/3,0,0,0,0,0,-(8/3),0,0,0,-4,2,0,0,0,-(8/3),0,0,0,-(8/3),2,0,0,0,2,2,0,0,0,0,-(8/3),0,-(8/3),0,0,-4,0,2}],AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{4,0,0,4/3,0,4/3,0,0,-(4/3),0,-(4/3),0,0,0,0,0,4/3,0,0,0,8/3,-(4/3),0,0,0,2,0,0,0,4/3,-(4/3),0,0,0,-(4/3),-(4/3),0,0,0,0,2,0,4/3,0,0,8/3,0,-(4/3)}],0,1}


One can readily verify, however, that this is not actually in the null space:

In[4] := M.n
Out[4] := {0,0,0,0,0,0,0,AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{-(38/3),0,0,-(28/3),0,-(16/3),0,0,16/3,0,28/3,0,0,0,0,0,-(28/3),0,0,0,-(38/3),16/3,0,0,0,-(34/3),0,0,0,-(28/3),14/3,0,0,0,16/3,14/3,0,0,0,0,-(34/3),0,-(28/3),0,0,-(38/3),0,16/3}],0,AlgebraicNumber[Root[1-#1+#1^5-#1^6+#1^10-#1^11+#1^13-#1^14+#1^15-#1^16+#1^18-#1^19+#1^20-#1^21+#1^23-#1^24+#1^25-#1^27+#1^28-#1^29+#1^30-#1^32+#1^33-#1^34+#1^35-#1^37+#1^38-#1^42+#1^43-#1^47+#1^48&,48],{-12,0,0,-(16/3),0,-(16/3),0,0,16/3,0,16/3,0,0,0,0,0,-(16/3),0,0,0,-(32/3),16/3,0,0,0,-6,0,0,0,-(16/3),16/3,0,0,0,16/3,16/3,0,0,0,0,-6,0,-(16/3),0,0,-(32/3),0,16/3}],0,0,0,0}


Can anyone explain what is going on? (Or produce a smaller example of this bug?)

• I'd also be curious to hear if other versions experience this problem. – Scott Morrison Nov 30 '14 at 23:40
• The 22 by 22 matrix you linked to has dimensions 14 by 14, and it is full rank. – bill s Nov 30 '14 at 23:41
• @bills; oops, I've corrected my claims, now! My original example had been block diagonal, and I cut it down to the relevant block, but forgot to update my post. – Scott Morrison Nov 30 '14 at 23:44
• I certainly agree the matrix is full rank. It's mathematica which seems to be having difficulty. :-) Which version are you running on? On 9.0.1.0, I'm definitely seeing rank 12 here. – Scott Morrison Nov 30 '14 at 23:44
• I"m on version 10.0.1 Mac os and get matrixrank 14. – bill s Nov 30 '14 at 23:45