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} } . 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. The principal invariants do not change with rotations of the coordinate system (they are objective, or in more modern terminology, satisfy the principle of material frame-indifference) and any function of the principal invariants is also objective.
Calculation of the invariants of rank two tensors In a majority of engineering applications, the principal invariants of (rank two) tensors of dimension three are sought, such as those for the right Cauchy-Green deformation tensor C {\displaystyle \mathbf {C} } which has the eigenvalues λ 1 2 {\displaystyle \lambda _{1}^{2}} , λ 2 2 {\displaystyle \lambda _{2}^{2}} , and λ 3 2 {\displaystyle \lambda _{3}^{2}} . Where λ 1 {\displaystyle \lambda _{1}} , λ 2 {\displaystyle \lambda _{2}} , and λ 3 {\displaystyle \lambda _{3}} are the principal stretches, i.e. the eigenvalues of U = C {\displaystyle \mathbf {U} ={\sqrt {\mathbf {C} }}} .
Principal invariants For such tensors, the principal invariants are given by:
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