make trace of a matrix listable?

Question

I have a list of 2 by 2 matrices and I want the list of their traces. I have been using

Map[Tr,mylistofmatrices]

But I wonder if there is a way to make Tr Listable so I can just call

Tr[mylistofmatrices]

Answer 1

No, there isn't. There are several reasons for that:

• Tr operates on tensors of arbitrary rank, not just matrices

• Listable functions will automatically thread to the deepest level of lists, so if you set Tr to be Listable, it'll individually wrap each deepest element of a nested list, e.g. Tr[{{1,2},{3,4}}] would transform to {{Tr[1], Tr[2]}, {Tr[3], Tr[4]}}.

Just use Map, which is unambiguous and takes just one extra character to type ... Tr /@ list vs Tr @ list.

Answer 2

If you have version 10 or later you can use operators forms, i.e. Map[Tr] acts as a function:

mylistofmatrices = {{{a, b}, {c, d}}, {{q, r, s}, {t, u, v}}};

Map[Tr][mylistofmatrices]

{a + d, q + u}

This is a subtle difference but it may be useful nevertheless.

If you always want the trace of a tensor with a specific array depth you can check for that. If you want only a two-dimensional tensor (which is what you state in your question) you can use MatrixQ; otherwise look at ArrayDepth and TensorRank. One implementation:

myTr[m_?MatrixQ] := Tr[m]
myTr[{m__?MatrixQ}] := Tr /@ {m}

Now:

myTr @ foo
myTr @ {1, 2, 3}
myTr @ {{a, b}, {c, d}}
myTr @ mylistofmatrices

myTr[foo]

myTr[{1, 2, 3}]

a + d

{a + d, q + u}