Invariants of tensors

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, in the fields of <a href="/facts/Multilinear_algebra/LrwpUFsM">multilinear algebra</a> and <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a>, the principal invariants of the second rank <a href="/facts/Tensor/pNjFo5tV">tensor</a> 
 
 
 
 
 A
 
 
 
 {\displaystyle \mathbf {A} }
 
 are the coefficients of the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a>

p
 (
 λ
 )
 =
 det
 (
 
 A
 
 −
 λ
 
 I
 
 )
 
 
 {\displaystyle \ p(\lambda )=\det(\mathbf {A} -\lambda \mathbf {I} )}
 
,
where 
 
 
 
 
 I
 
 
 
 {\displaystyle \mathbf {I} }
 
 is the identity operator and 
 
 
 
 
 λ
 
 i
 
 
 ∈
 
 C
 
 
 
 {\displaystyle \lambda _{i}\in \mathbb {C} }
 
 are the roots of the polynomial 
 
 
 
  
 p
 
 
 {\displaystyle \ p}
 
 and the <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> of 
 
 
 
 
 A
 
 
 
 {\displaystyle \mathbf {A} }
 
.
More broadly, any scalar-valued function 
 
 
 
 f
 (
 
 A
 
 )
 
 
 {\displaystyle f(\mathbf {A} )}
 
 is an invariant of 
 
 
 
 
 A
 
 
 
 {\displaystyle \mathbf {A} }
 
 if and only if 
 
 
 
 f
 (
 
 Q
 
 
 A
 
 
 
 Q
 
 
 T
 
 
 )
 =
 f
 (
 
 A
 
 )
 
 
 {\displaystyle f(\mathbf {Q} \mathbf {A} \mathbf {Q} ^{T})=f(\mathbf {A} )}
 
 for all orthogonal 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbf {Q} }
 
. This means that a formula expressing an invariant in terms of components, 
 
 
 
 
 A
 
 i
 j
 
 
 
 
 {\displaystyle A_{ij}}
 
, will give the same result for all Cartesian bases. For example, even though individual diagonal components of 
 
 
 
 
 A
 
 
 
 {\displaystyle \mathbf {A} }
 
 will change with a change in basis, the sum of diagonal components will not change.

Invariants of tensors open-in-new

Invariants of tensors