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I've defined three symbolic abstract matrices X, M and S as shown below.

In[1]:= $Assumptions = { 
Element[X, Matrices[{k, 1}]],
Element[M, Matrices[{k, 1}]],
Element[S, Matrices[{k, k}, Reals, Symmetric[{1, 2}]]]
};

In[7]:= prodA = TensorTranspose[X].S.M;
        prodB = TensorTranspose[M].S.X;

In[12]:= TensorReduce[prodA + prodB]
Out[12]= TensorTranspose[M, {2, 1}].S.X + TensorTranspose[X, {2, 1}].S.M

Since the S is a square and symmetric matrix, the production results prodA and prodB should be equal. How can I let the output reflect that? I need the output show something like 2TensorTranspose[X].S.M.

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You can replace Matrices[{k, 1}] to Vectors[k]

$Assumptions = {Element[X, Vectors[k]], Element[M, Vectors[k]], 
   Element[S, Matrices[{k, k}, Reals, Symmetric[{1, 2}]]]};

prodA = X.S.M; (* vector not needed to be transposed *)
prodB = M.S.X;

TensorReduce[prodA + prodB]
2 M.S.X

TensorReduce can't simplify Matrices[{k, 1}] because your relation is not true for general dimensions:

u = RandomReal[1.0, {10, 2}];
v = RandomReal[1.0, {10, 2}];
m = (# + Transpose[#]) &@RandomReal[1.0, {10, 10}];

Transpose[u].m.v // MatrixForm
Transpose[v].m.u // MatrixForm

enter image description here

enter image description here

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  • $\begingroup$ Exactly, using Vector declaration solves the problem. You're absolutely right! Thank you! $\endgroup$ – Qi Qi Oct 14 '13 at 12:11

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