Invariants of tensors

In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor ${\displaystyle \mathbf {A} }$ are the coefficients of the characteristic polynomial^[1]

${\displaystyle \ p(\lambda )=\det(\mathbf {A} -\lambda \mathbf {I} )}$,

where ${\displaystyle \mathbf {I} }$ is the identity operator and ${\displaystyle \lambda _{i}\in \mathbb {C} }$ are the roots of the polynomial ${\displaystyle \ p}$ and the eigenvalues of ${\displaystyle \mathbf {A} }$.

More broadly,any scalar-valued function ${\displaystyle f(\mathbf {A} )}$ is an invariant of ${\displaystyle \mathbf {A} }$ if and only if ${\displaystyle f(\mathbf {Q} \mathbf {A} \mathbf {Q} ^{T})=f(\mathbf {A} )}$ for all orthogonal ${\displaystyle \mathbf {Q} }$. This means that a formula expressing an invariant in terms of components, ${\displaystyle A_{ij}}$, will give the same result for all Cartesian bases. For example, even though individual diagonal components of ${\displaystyle \mathbf {A} }$ will change with a change in basis, the sum of diagonal components will not change.