Invariants of Tensors

In mathematics, in the fields of multilinear algebra and representation theory, invariants of tensors are coefficients of the characteristic polynomial of the tensor A:

,

where is the identity tensor and is the polynomial's indeterminate (it is important to bear in mind that a polynomial's indeterminate may also be a non-scalar as long as power, scaling and adding make sense for it, e.g., is legitimate, and in fact, quite useful).

The first invariant of an n×n tensor A is the coefficient for (coefficient for is always 1), the second invariant is the coefficient for, etc., the nth invariant is the free term.

The definition of the invariants of tensors and specific notations used throughout the article were introduced into the field of Rheology by Ronald Rivlin and became extremely popular there. In fact even the trace of a tensor is usually denoted as in the textbooks on rheology.