“Trace” is a particular operator that, when “applied” to a order (or higher) tensor, sums the diagonal components. Strictly-speaking, the definition of “trace” can be taken as

unless the tensor is symmetric

The “scalar product” of tensors is analogous to the “dot product” of vectors. In this text, the scalar product is denoted by the operator “:” and is defined as follows:

We can arrive at the same result in a different manner:

note: Since the scalar product of two tensors is analogous to the “dot product” of two vectors, some authors define the scalar product between and as [Hoger]. In other words, these authors define . Using such a definition, , which still must be equal to , can no longer be expressed as , as was done previously.

An eigenvector, , of a tensor or a matrix has the following special property:

when is multiplied by the matrix, the result is a new vector that has the same direction as . The amount by which the magnitude of the vector has changed is the value of the *eigenvalue* (the eigenvalue that corresponds to the eigenvector ).

*i.e.*

are the eigenvalue solutions and are the corresponding eigenvectors

Any tensor product can be expressed in terms of invariants (or eigenvalues)

*i.e.*

(skipped work)

or

Characteristic equation and Cayley-Hamilton Theorem:

One would typically solve for the eigenvalues from a “characteristic equation” of the form (for a 3×3 matrix), where , , are coefficients that depend on the values within the matrix, . These coefficients, , , , are more commonly called invariants.

We’ll see some alternative expressions for , , later, when we get to “hyperelasticity.”

note: The invariants of a matrix are the same regardless of coordinate system (as is the trace).

The matrix also satisfies its own characteristic equation (this is known as the Cayley-Hamilton Theorem).

*i.e.*

(1)

(2)

Either eq. 1 or eq. 2 are often used in derivations of hyperelasticity later on.