In Wolfram|Alpha it is possible to discover some specific classes to which a matrix belongs.
For instance, typing this in Wolfram|Alpha
{{a, b, c}, {c, a, b}, {b, c, a}}
returns
Properties: normal circulant
Is there a built-in Mathematica command which can generate the same kind of output?
As far is I can tell, there is no built-in command to do so. However, typing
?*MatrixQ
Into the front end and evaluating will give a list of the matrix properties you can test for in Mathematica. Based on the resulting list, I came up with the following simple function that you might find useful.
This is just a helper function to make the output look nice:
MatrixQtoTitle[a_] :=
StringTrim[StringReplace[ ToString[a], {"MatrixQ" -> "",
uc : CharacterRange["A", "Z"] :> " " ~~ ToLowerCase[uc]}]]
And here is the suggested function, using the output from the ?*MatrixQ line mentioned above:
MatrixPropertiesReport[mat_?MatrixQ] :=
Grid[{MatrixQtoTitle[#], #[mat]} & /@ {AntihermitianMatrixQ,
AntisymmetricMatrixQ, DiagonalizableMatrixQ, HermitianMatrixQ,
IndefiniteMatrixQ, NegativeDefiniteMatrixQ,
NegativeSemidefiniteMatrixQ, NormalMatrixQ, OrthogonalMatrixQ,
PositiveDefiniteMatrixQ, PositiveSemidefiniteMatrixQ,
SquareMatrixQ, SymmetricMatrixQ, UnitaryMatrixQ}]
Using your example
antihermitian False
antisymmetric False
diagonalizable True
hermitian False
indefinite False
negative definite False
negative semidefinite False
normal True
orthogonal False
positive definite False
positive semidefinite False
square True
symmetric False
unitary False
If you wanted you could adapt this function to, for example, only show the rows that are true, much as Wolfram|Alpha does.
MatrixPropertiesReport[mat_?MatrixQ] := If[ToExpression[#][mat], ToLowerCase[Fold[StringReplace, #, {"MatrixQ" -> "", x_?LowerCaseQ ~~ y_?UpperCaseQ :> x <> " " <> y}]], Nothing] & /@ DeleteCases[Names["*MatrixQ"], "MatrixQ"]
$\endgroup$