4.2. Determinants

Det            Determinant of a tensor of arbitrary rank

Determinants

We can define the determinant of a tensor in any given basis through the η tensors:

In[240]:=

Det[T[a, b], polar]

Out[240]=

1/6 Underscript[η, ~] _ace^    Underscript[η, ~] _bdf^    T_  ^ab T_  ^cd T_  ^ef

In[241]:=

WeightOf[%]

Out[241]=

-2 polar

Notice how the determinant depends on the basis, unless there are the same number of contravariant and covariant indices.

In[242]:=

DefTensor[T2[a, -b], M3, PrintAs→ "T"]

** DefTensor: Defining tensor T2[a, -b] .

In[243]:=

Det[T2[a, -b], polar]

Out[243]=

1/6 δ_ (   ace)^bdf    T_ ( b)^a T_ ( d)^c T_ ( f)^e

In[244]:=

WeightOf[%]

Out[244]=

0

If a tensor has an odd number of indices, its determinant is zero

In[245]:=

DefTensor[T3[a, b, c], M3, PrintAs→ "T"]

** DefTensor: Defining tensor T3[a, b, c] .

In[246]:=

Det[T3[a, b, c], cartesian]

Out[246]=

In[247]:=

ToCanonical[%]

Out[247]=

0

In[248]:=

DefTensor[T4[a, b, c, -d], M3, PrintAs→ "T"]

** DefTensor: Defining tensor T4[a, b, c, -d] .

In[249]:=

Det[T4[a, b, c, -d], cartesian]//ToCanonical

Out[249]=

Even if we know that the determinant of a given object is going to be basis independent, we must provide some basis:

In[250]:=

{Det[delta[a, -b]], Det[delta[a, -b], polar]}

Out[250]=

{Det[δ_ ( b)^a ], 1}

In[251]:=

UndefTensor /@ {T2, T3, T4} ;

** UndefTensor: Undefined tensor T2

** UndefTensor: Undefined tensor T3

** UndefTensor: Undefined tensor T4

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