10

So, I'd like to define a matrix M, that does not decompose into it's constituents when I do things like Tr[M], but I also want it's type to be retained.

(By type, I mean, what sort of a data structure it is, integer, or matrix, or vector, not literally the data-types available.)

For example:

In: Tr[a*M]

Out: a*Tr[M]

The solution from this answer doesn't seem to retain the type.

Is there any way to do this?

Edit: Basically, have an atomic symbol whose type is say a 2*3 matrix (or whatever is the equivalent of that in Mathematica), but whose individual elements don't matter for computation.

If I define a matrix as follows:

M = {{a,b},{c,d}}

Then, Tr[M] gives a+d. But I don't want the symbol to expand that way.

I'd like Tr[M] to give Tr[M] but Tr[a*M] to give a*Tr[M] and have other linear functions expand similarly.

Carl Woll
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Mahathi Vempati
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  • I do not understand the question (not even what you mean by "type") ... a lot more explanation is needed, with concrete examples and a clear explanation of the motivation. Please edit and update the question. – Szabolcs Jun 16 '19 at 10:54
  • "have an atomic symbol" makes no sense because all symbols are atomic. {{a,b},{c,d}} is not atomic, and is not a symbol. Thus, do not give a value to M, keep it as a symbol, and look up symbolic tensor manipulations in the docs. Perhaps this is what you want. – Szabolcs Jun 16 '19 at 11:25

2 Answers2

11

Perhaps what you want is symbolic tensors: http://reference.wolfram.com/language/tutorial/SymbolicTensors.html

$Assumptions = M ∈ Arrays[{3, 3}] && a ∈ Complexes;

TensorContract[a*M, {{1, 2}}]
(* a TensorContract[M, {{1, 2}}] *)
Szabolcs
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  • Thank you very much! How come this doesn't work with the function Tr, since it's just a specific case of TensorContract? – Mahathi Vempati Jun 16 '19 at 12:50
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    @MahathiVempati I don't know, I'm not experienced in this area. But TensorContract was introduced specifically for symbolic tensor manipulations. Not every function supports this functionality. – Szabolcs Jun 16 '19 at 13:10
11

You could use my TensorSimplify package help with this. Install the paclet with:

PacletInstall[
    "TensorSimplify", 
    "Site" -> "http://raw.githubusercontent.com/carlwoll/TensorSimplify/master"
]

Once installed, you can load the package with:

<<TensorSimplify`

This package defines the helper functions ToTensor and FromTensor. ToTensor converts expressions containing Tr and Dot into the equivalent version using TensorContract, and FromTensor does the opposite. So, your Tr expression can be simplified with:

$Assumptions = M \[Element] Matrices[{3, 3}] && a \[Element] Complexes;

FromTensor @ ToTensor @ Tr[a M]

a Tr[M]

An alternative is to just use the package's TensorSimplify function:

TensorSimplify @ Tr[a M]

a Tr[M]

TensorSimplify will work for more complicated arguments to Tr:

TensorSimplify @ Tr[a M + a^2 M.M]

a Tr[M] + a^2 Tr[M.M]

Carl Woll
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