❤: Anisotropic
# AN EIGENANALYSIS OF $I_5$
## The Eigensystem of $I_5$
We will now show that the eigensystem of any energy expressed solely in terms of $I_5$ can be written down in closed form. The $I_5$ invariant can be written in several forms,
``` iheartla
tr from linearalgebra
`$I_5$` = tr(CA)
where
a ∈ ℝ^3
C ∈ ℝ^(3×3)
```
where ❤ A = a a^T ❤ and $\|\cdot\|_{2}^{2}$ denotes the squared Euclidean norm. The PK1 and Hessian in 3D are
``` iheartla
`$\frac{∂I₅}{∂F}$` = 2FA
F ∈ ℝ^(3×3): scaling and rotation matrix
```
``` iheartla
`$\frac{∂²I₅}{∂f²}$` = 2[A₁,₁I₃ A₁,₂I₃ A₁,₃I₃
A₂,₁I₃ A₂,₂I₃ A₂,₃I₃
A₃,₁I₃ A₃,₂I₃ A₃,₃I₃]
```
where $I _{3×3}$ is a 3×3 identity matrix,and $A _{ij}$ is the $(i, j)$ scalar entry of $A$. (Appendix A shows the matrix explicitly.) Since Eqn. 7 is constant in $a$, it is straightforward to state its eigensystem in closed form. In 3D, it contains three identical non-zero eigenvalues, ❤`$\lambda_{0,1,2}$`=2||a||_2^2❤, and since fiber directions are usually normalized, this simplifies to $\lambda_{0,1,2}=2$. The eigenvalue is repeated, so the eigenmatrices are arbitrary up to rotation, but one convenient phrasing is:
``` iheartla_unnumbered
`$\mathbf{Q}_{0}$` = [a^T
0
0]
`$\mathbf{Q}_{1}$` = [0
a^T
0]
`$\mathbf{Q}_{2}$` = [ 0
0
a^T]
```
This eigenstructure has a straightforward interpretation. $I_5$ introduces scaling constraints along the anisotropy direction, so the three eigenvectors encode this rank-three phenomenon. The remaining eigenvalues are all zero, so the Hessian contains a rank-six null space. We have provided supplemental Matlab/Octave code that validate these expressions.
❤: Anisotropic2D
The 2D case follows similarly. The Hessian is
``` iheartla
tr from linearalgebra
`$\frac{∂²I₅}{∂f²}$` = 2[A₁,₁I_2 A₁,₂I_2
A₂,₁I_2 A₂,₂I_2]
where
a ∈ ℝ^2
A ∈ ℝ^(2×2)
```
the eigenvalues are ❤`$\lambda_{0,1}$`=2||a||_2^2❤, and the eigenmatrices become
``` iheartla_unnumbered
`$\mathbf{Q}_{0}$` = [a^T
0]
`$\mathbf{Q}_{1}$` = [0
a^T]
```