Trace, Scalar Product, Eigenvalues

“Trace” is a particular operator that, when “applied” to a $2^{nd}$ order (or higher) tensor, sums the diagonal components. Strictly-speaking, the definition of “trace” can be taken as $tr(\mathbf{u}\otimes\mathbf{v})=\mathbf{u\cdot v}$

$tr(\mathbf{A})=A_{ii}$
$tr(\mathbf{A \cdot B})=tr(\mathbf{C})=C_{ii}$
$C_{ij}=A_{ik}B_{kj} \ne A_{ik}B_{jk}$ unless the tensor is symmetric
$tr(\mathbf{A \cdot B})=C_{ii}=A_{ik}B_{ki}$
$tr(\mathbf{A \cdot B \cdot C})=tr(\mathbf{B \cdot C \cdot A})=tr(\mathbf{C \cdot A \cdot B})=A_{ik}B_{kl}C_{li}$

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:

$\mathbf{A:B}=tr(\mathbf{A}^{T} \cdot \mathbf{B})=tr(\mathbf{A} \cdot \mathbf{B}^T)=A_{ik}B_{ik}$

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

$\mathbf{A:B}=A_{ij}\mathbf{e_ie_j}:B_{lk}\mathbf{e_l e_k}=A_{ij}B_{lk}\delta_{il}\delta_{jk}=A_{ik}B_{ik}$

note: Since the scalar product of two tensors is analogous to the “dot product” of two vectors, some authors define the scalar product between $\mathbf{A}$ and $\mathbf{B}$ as $\mathbf{A \cdot B}$ [Hoger]. In other words, these authors define $\mathbf{A} \cdot \mathbf{B}=A_{ik}B_{ik}$ . Using such a definition, $\mathbf{A \cdot B}$ , which still must be equal to $A_{ij}\mathbf{e_ie_j} \cdot B_{kl}\mathbf{e_ke_l}$ , can no longer be expressed as $A_{ij}B_{kl}\delta_{jk}\mathbf{e_i}\mathbf{e_l}$ , as was done previously.

An eigenvector, $\mathbf{n}$ , of a tensor or a matrix has the following special property:
when $\mathbf{n}$ is multiplied by the matrix, the result is a new vector that has the same direction as $\mathbf{n}$ . The amount by which the magnitude of the vector has changed is the value of the eigenvalue (the eigenvalue that corresponds to the eigenvector $\mathbf{n}$ ).

i.e. $(\mathbf{A}-\lambda\mathbf{I}) \cdot \mathbf{n}=0$

$\lambda$ are the eigenvalue solutions and $\mathbf{n}$ are the corresponding eigenvectors
Any tensor product can be expressed in terms of invariants (or eigenvalues)
i.e.
$\boldsymbol{\epsilon}=[\lambda_{1}n_{i}^{1}n_{j}^{1}+\lambda_{2}n_{i}^{2}n_{j}^{2}+\lambda_{3}n_{i}^{3}n_{j}^{3}]\mathbf{e_i \otimes e_j}$ (skipped work)
or
$\boldsymbol{\epsilon}=\sum_{a=1}^{3}\lambda_{a}\mathbf{n_a \otimes n_a}$

Characteristic equation and Cayley-Hamilton Theorem:

One would typically solve for the eigenvalues from a “characteristic equation” of the form $\lambda^3-I_A\lambda^2+II_A\lambda-III_A=0$ (for a 3×3 matrix), where $I_A$ , $II_A$ , $III_A$ are coefficients that depend on the values within the matrix, $\mathbf{A}$ . These coefficients, $I_A$ , $II_A$ , $III_A$ , are more commonly called invariants.

$I_A=\lambda_1+\lambda_2+\lambda_3$
$II_A=\lambda_1\lambda_2+\lambda_2\lambda_3+\lambda_3\lambda_1$
$III_A=\lambda_1\lambda_2\lambda_3$

We’ll see some alternative expressions for $I_A$ , $II_A$ , $III_A$ 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) $\begin{equation*} \mathbf{A}^3-I_A\mathbf{A}^2+II_A\mathbf{A}-III_A\mathbf{I}=0 \end{equation*}$

$\mathbf{A}^{-1}(\mathbf{A}^3-I_A\mathbf{A}^2+II_A\mathbf{A})=III_A\mathbf{I}\mathbf{A}^{-1}$
$\mathbf{A}^2-I_A\mathbf{A}+II_A\mathbf{I}=III_A\mathbf{A}^{-1}$

(2) $\begin{equation*} \frac{1}{III_A}(\mathbf{A}^2-I_A\mathbf{A}+II_A\mathbf{I})=\mathbf{A}^{-1} \end{equation*}$

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

A. Hoger, “A second order constitutive theory for hyperelastic materials,” International Journal of Solids and Structures, vol. 36, iss. 6, pp. 847-868, 1999.
[Bibtex]

@article{Hoger,
title = "A second order constitutive theory for hyperelastic materials",
journal = "International Journal of Solids and Structures",
volume = "36",
number = "6",
pages = "847 - 868",
year = "1999",
note = "",
issn = "0020-7683",
doi = "10.1016/S0020-7683(97)00330-2",
url = "http://www.sciencedirect.com/science/article/pii/S0020768397003302",
author = "Anne Hoger"
}

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