How to solve a "tensor equation"?

Question

I am trying to solve equations which looks like this:

$$ T_{ab} - T_{bc} = a_1 T_{ab} + a_2 T_{ac} + a_3 T_{bc}, $$

where $T_{xy}$ are tensors. I want to get the $a_i$'s (in this simple example $a_1=1$, $a_2=0$, $a_3=-1$).

The problem is that the Mathematica solving routines (Solve, LinearSolve) divide the equation by $T_{xy}$ to obtain a solution (or solutions), which is (or some of them are), in turn, not a solution (solutions).

What I tried to circumvent this problem: I know (from the construction of my equation) that all $a_i$'s are element $\{-1,0,1\}$. So I tried to set the domain to integers. But this gives me lots of condidtional solutions (like if Tab=integer, then...) which I don't want.

Of course I could solve the above example equation by hand, but in the end I will need to solve a few hundred equations of this type with 70 tensors or so.

Thanks in advance for any attempt to help me!

Anton

You appear to misunderstand basic Mathematica syntax. Start learning documentation on the new tensor functionality in ver.9. For earlier versions this post might be helpful: How to calculate scalar curvature Ricci tensor and Christoffel symbols in Mathematica?. For solving matrix equations see e.g. General form of a linear transformation. — Artes, Nov 19 '13 at 10:53
No, I do not misunderstand basic Mathematica syntax. But, what you couldn't know: I am working with an older special package that forces me to use Mathematica 8 and a special syntax. (And I necessarily need that package.) — user201018, Nov 19 '13 at 12:03
Really ? So why don't you use correct Mathematica notation, but something like a_1, a_2 etc? — Artes, Nov 19 '13 at 12:22
In my Mathematica code I write T[a,b] and a[i] etc. I just wanted to give an as simple as possible example that illustrates my problem. In the future, I'll not change the notation, or should I? — user201018, Nov 19 '13 at 13:30
You should always provide valid Mathematica code. So if you have different notation in your notebook you can edit your question to make it correct. By the way I didn't vote down your question, I find questions on tensors interesting. — Artes, Nov 19 '13 at 14:38

score 5 · Accepted Answer · answered Nov 19 '13 at 11:00

5

I don't think you need to use any specific tensor functionality. SolveAlways seems to suffice:

SolveAlways[ T1 - T3 == a1 T1 + a2 T2 + a3 T3, {T1, T1, T3} ]
(* => {{a1 -> 1, a2 -> 0, a3 -> -1}} *)

answered Nov 19 '13 at 11:00

Teake Nutma

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This solution works fine unless you have large equations. Then Mathematica needs an tremendous amount of memory. So you need to "help" Mathematica... the following worked for me (basically, I split up the "tensor" eq into one equation for each tensor): – user201018 Nov 25 '13 at 10:19
I created a new answer due to the length limit for the comments. – user201018 Nov 25 '13 at 10:26

user201018 · Answer 2 · 2013-11-25T11:46:01.797

please read the comment below the answer that I marked as solved

 (* put all the terms to one side of the eq *)
tensorEq = -(T1 - T3) + 
  a1 T1 + a2 T2 + a3 T3
(* collect the terms that belong together *)
tensorTermsList = 
 Apply[List, Collect[tensorEq, {T1, T2, T3}]]
(* set the terms individually to 0 *)
tensorEqsList = 
 Table[tensorTermsList[[i]] == 0, {i, 1, Length[tensorTermsList]}]
(* now we can give the full list to SolveAlways, SolveAlways can 
handle a large number of simple equations *)
sol = 
 SolveAlways[tensorEqsList, {T1, T2, T3}][[1]]

How to solve a "tensor equation"?

2 Answers2