Invariants of tensors
In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor A {\displaystyle \mathbf {A} } are the coefficients of the characteristic polynomial 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 eigenvalues 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} } .