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Bug introduced in 10.0 and fixed in 10.2.0


I found a case where calling one built-in function (HermitianMatrixQ) for a given matrix changes the behavior of another built-in function (AntihermitianMatrixQ) for the same matrix object. Here is the code:

M = {{0, 1}, {-1, 0}}

AntihermitianMatrixQ[M]    (* output: True *)
HermitianMatrixQ[M]        (* output: False *)
AntihermitianMatrixQ[M]    (* output: False -- WRONG! *)

I observed this in Mathematica 10.0.0.0 on Microsoft Windows (64-bit).

The behavior is "stateful" in the sense that re-evaluating the assignment to M restores the correct behavior of AntihermitianMatrixQ until the next time the function HermitianMatrixQ is called for the same object, from which point on AntihermitianMatrixQ gives the incorrect result again.

I did some experimentation and found the following:

  • The "state" is per object: When you define several array variables, the functions behave independently for each of the array objects, in the way described above.
  • The problem does not occur for all matrices, but it is not limited to the given 2x2 case, either. (In fact I found it when debugging a calculation with 4x4 Dirac gamma matrices.)
  • The function SymmetricMatrixQ intereferes with the behavior of AntihermitianMatrixQ in the same way.
  • The result of AntisymmetricMatrixQ is affected in the same way.
  • Printing the FullForm of the object does not give any indication of a change to the object itself.

My question is therefore:

  • Does Mathematica store any per-object data for matrices that is not displayed in FullForm but that can affect the behavior of functions as described above?
  • If so, is there any way to display or to clear this kind of additional data?

Note: I created a support request with Wolfram Research regarding this strange behavior. So far I got no response, however. I would also appreciate independent confirmation of this behavior, to exclude the possibility that something is wrong with my particular installation of Mathematica.

UPDATE: I got a response to my support request (CASE:3202743) from WRI. They confirm that AntisymmetricMatrixQ is not behaving properly and that an incident report has been created. Many thanks also to @ilian for pursuing this matter. I would still be interested in answers to my questions as formulated -- both out of curiosity and in order to improve debugging options.

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    $\begingroup$ I see the same behavior on OS X MMA ver.10.0.1.0 $\endgroup$ – N.J.Evans May 7 '15 at 21:29
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    $\begingroup$ Thank you, I've submitted a bug report to the appropriate developers. $\endgroup$ – ilian May 7 '15 at 21:45
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    $\begingroup$ I see the same behavior in MMA ver. 10.1.0 on Win7-64bit. This is weird! $\endgroup$ – MarcoB May 7 '15 at 21:45
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    $\begingroup$ I think a workaround might be to use an explicit SameTest. {AntihermitianMatrixQ[m, SameTest -> (Simplify[#1 - #2] == 0 &)], HermitianMatrixQ[m, SameTest -> (Simplify[#1 - #2] == 0 &)], AntihermitianMatrixQ[m, SameTest -> (Simplify[#1 - #2] == 0 &)]} $\endgroup$ – chuy May 7 '15 at 22:16
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    $\begingroup$ Indeed there is something certainly amiss. See this: f = (Print[Equal[##]]; Equal[##]) &; AntihermitianMatrixQ[m, SameTest -> f] as opposed to HermitianMatrixQ[m, SameTest -> f]. $\endgroup$ – chuy May 7 '15 at 22:30
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Now fixed in version 10.2.

In[1]:= m = {{0, 1}, {-1, 0}};

In[2]:= {AntihermitianMatrixQ[m], HermitianMatrixQ[m], AntihermitianMatrixQ[m]} 

Out[2]= {True, False, True}

As per the comments, yes, there is information stored in the internal representation of matrices (for example, a symmetry flag) and no, it is not accessible from top level code.

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