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I managed to to it relatively simply. This introduces the objects: $Assumptions = {Element[x, Matrices[{3, 1}, Reals]], Element[a, Matrices[{3, 1}, Reals]], Element[n, Matrices[{3, 1}, Reals]], Element[b, Matrices[{3, 1}, Reals]], Element[m, Matrices[{3, 1}, Reals]], Element[id, Matrices[{3, 3}, Reals, Symmetric]]}; and this set of ...


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You cannot use @ as it is used for assignments etc. So just set up your own tensor product tp and inner product ip and define tp[id,v_]:=v tp[v_,id]:=v Tra[tp[a,b]]:=ip[a,b] Trans[tp[a,b]]:=tp[b,a] ip[tp[a,b],tp[c,d]]:=ip[b,c] tp[a,d] and tp[a_+b_,c_]:=tp[a,c]+tp[b,c] It is unclear as to what you mean by "respect scalar multiplication". If you only ...


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I think the closest you can get to what you looking for is Assuming[{a, b, c} ∈ Reals, With[{x = {a, b, c}}, x.x == Norm[x]^2 // Simplify]] True


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As suggested already in the comments it is easier to work in a concrete basis e.g. as follows: x = {1, 0} ; y = {0, 1}; bv[a_, b_] := Flatten[KroneckerProduct[a, b]]; Then you can calculate your particular examples easily: (a*bv[x, x] + c bv[y, x]).(b bv[x, y]) (* 0 *) (a*bv[x, x] + c bv[x, y]).(b bv[x, x]) (* a*b *) Using the symbolic tensors in ...


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For example, << xAct`xPert` Define your structures: DefManifold[M4, 4, IndexRange[a, f]] DefMetric[{1, 3, 0}, g[-a, -b], CD] DefMetricPerturbation[g, H, \[Epsilon]] Construct the object to perturb, as you give it above, including the epsilon tensor of the metric g: GR = 1/2 epsilong[a, b, c, d] RiemannCD[e, -f, -c, -d] RiemannCD[f, -e, -a, -b] ...



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