In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor
More broadly, any scalar-valued function
This means that a formula expressing an invariant in terms of components,
, will give the same result for all Cartesian bases.
For example, even though individual diagonal components of
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.
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
[2] The correspondence between the principal invariants and the characteristic polynomial of a tensor, in tandem with the Cayley–Hamilton theorem reveals that where
is the second-order identity tensor.
These are the coefficients of the characteristic polynomial of the deviator
The separation of a tensor into a component that is a multiple of the identity and a traceless component is standard in hydrodynamics, where the former is called isotropic, providing the modified pressure, and the latter is called deviatoric, providing shear effects.
Furthermore, mixed invariants between pairs of rank two tensors may also be defined.
[4] These may be extracted by evaluating the characteristic polynomial directly, using the Faddeev-LeVerrier algorithm for example.
The invariants of rank three, four, and higher order tensors may also be determined.
that depends entirely on the principal invariants of a tensor is objective, i.e., independent of rotations of the coordinate system.
This property is commonly used in formulating closed-form expressions for the strain energy density, or Helmholtz free energy, of a nonlinear material possessing isotropic symmetry.
[6] This technique was first introduced into isotropic turbulence by Howard P. Robertson in 1940 where he was able to derive Kármán–Howarth equation from the invariant principle.
[7] George Batchelor and Subrahmanyan Chandrasekhar exploited this technique and developed an extended treatment for axisymmetric turbulence.
in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part.
are the first step would be to evaluate the axial vector
Specifically, the axial vector has components The next step finds the principal values of the symmetric part of
Even though the eigenvalues of a real non-symmetric tensor might be complex, the eigenvalues of its symmetric part will always be real and therefore can be ordered from largest to smallest.
The corresponding orthonormal principal basis directions can be assigned senses to ensure that the axial vector
With respect to that special basis, the components of
are the diagonal components of this matrix:
(equal to the ordered principal values of the tensor's symmetric part).
The remaining three invariants are the axial vector's components in this basis:
Note: the magnitude of the axial vector,
, is the sole invariant of the skew part of
, whereas these distinct three invariants characterize (in a sense) "alignment" between the symmetric and skew parts of