$x = \{$ $x$, $y$, $z$ $\}$
${{ \beta} ^i} = {\overset{i\downarrow}{\left[\begin{matrix} {\beta^x} \\ {\beta^y} \\ {\beta^z}\end{matrix}\right]}}$

---

${{ A} _x} = {\left[\begin{array}{cccc} -{{\beta^x}}& {-{\alpha}} {{{\gamma^{xx}}}}& {-{\alpha}} {{{\gamma^{xy}}}}& {-{\alpha}} {{{\gamma^{xz}}}}\\ -{\alpha}& -{{\beta^x}}& 0& 0\\ 0& 0& -{{\beta^x}}& 0\\ 0& 0& 0& -{{\beta^x}}\end{array}\right]}$
char poly:
${{{{\left({{{\beta^x}} + {\lambda}}\right)}^{2}}} {{\left({{{\left({{{\beta^x}} + {\lambda}}\right)}^{2}}{-{{{{\gamma^{xx}}}} \cdot {{{\alpha}^{2}}}}}}\right)}}} = {0}$
lambdas ${-{{\beta^x}}}{-{{{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}}}}$, $-{{\beta^x}}$, ${-{{\beta^x}}} + {{{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}}}$
for eigenvalue ${\lambda} = {{-{{\beta^x}}}{-{{{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}}}}}$
eigenvectors are
$\left[\begin{array}{c} \sqrt{{\gamma^{xx}}}\\ 1\\ 0\\ 0\end{array}\right]$
for eigenvalue ${\lambda} = {-{{\beta^x}}}$
eigenvectors are
$\left[\begin{array}{cc} 0& 0\\ -{{\frac{1}{{\gamma^{xx}}}} {{\gamma^{xy}}}}& -{{\frac{1}{{\gamma^{xx}}}} {{\gamma^{xz}}}}\\ 1& 0\\ 0& 1\end{array}\right]$
for eigenvalue ${\lambda} = {{-{{\beta^x}}} + {{{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}}}}$
eigenvectors are
$\left[\begin{array}{c} -{\sqrt{{\gamma^{xx}}}}\\ 1\\ 0\\ 0\end{array}\right]$
multiplicities 1, 2, 1
$\Lambda$:
$\left[\begin{array}{cccc} {-{{\beta^x}}}{-{{{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}}}}& 0& 0& 0\\ 0& -{{\beta^x}}& 0& 0\\ 0& 0& -{{\beta^x}}& 0\\ 0& 0& 0& {-{{\beta^x}}} + {{{\alpha}} \cdot {{\sqrt{{\gamma^{xx}}}}}}\end{array}\right]$
R:
$\left[\begin{array}{cccc} \sqrt{{\gamma^{xx}}}& 0& 0& -{\sqrt{{\gamma^{xx}}}}\\ 1& -{{\gamma^{xy}}}& -{{\gamma^{xz}}}& 1\\ 0& {\gamma^{xx}}& 0& 0\\ 0& 0& {\gamma^{xx}}& 0\end{array}\right]$
L:
$\left[\begin{array}{cccc} \frac{1}{{{2}} {{\sqrt{{\gamma^{xx}}}}}}& \frac{1}{2}& \frac{{\gamma^{xy}}}{{{2}} {{{\gamma^{xx}}}}}& \frac{{\gamma^{xz}}}{{{2}} {{{\gamma^{xx}}}}}\\ 0& 0& \frac{1}{{\gamma^{xx}}}& 0\\ 0& 0& 0& \frac{1}{{\gamma^{xx}}}\\ -{\frac{1}{{{2}} {{\sqrt{{\gamma^{xx}}}}}}}& \frac{1}{2}& \frac{{\gamma^{xy}}}{{{2}} {{{\gamma^{xx}}}}}& \frac{{\gamma^{xz}}}{{{2}} {{{\gamma^{xx}}}}}\end{array}\right]$
A check: $\left[\begin{array}{cccc} -{{\beta^x}}& -{{{\alpha}} \cdot {{{\gamma^{xx}}}}}& -{{{\alpha}} \cdot {{{\gamma^{xy}}}}}& -{{{\alpha}} \cdot {{{\gamma^{xz}}}}}\\ -{\alpha}& -{{\beta^x}}& 0& 0\\ 0& 0& -{{\beta^x}}& 0\\ 0& 0& 0& -{{\beta^x}}\end{array}\right]$ A diff: $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

---

${{ A} _y} = {\left[\begin{array}{cccc} -{{\beta^y}}& {-{\alpha}} {{{\gamma^{yx}}}}& {-{\alpha}} {{{\gamma^{yy}}}}& {-{\alpha}} {{{\gamma^{yz}}}}\\ 0& -{{\beta^y}}& 0& 0\\ -{\alpha}& 0& -{{\beta^y}}& 0\\ 0& 0& 0& -{{\beta^y}}\end{array}\right]}$
char poly:
${{{{\left({{{\beta^y}} + {\lambda}}\right)}^{2}}} {{\left({{{\left({{{\beta^y}} + {\lambda}}\right)}^{2}}{-{{{{\gamma^{yy}}}} \cdot {{{\alpha}^{2}}}}}}\right)}}} = {0}$
lambdas ${-{{\beta^y}}}{-{{{\alpha}} \cdot {{\sqrt{{\gamma^{yy}}}}}}}$, $-{{\beta^y}}$, ${-{{\beta^y}}} + {{{\alpha}} \cdot {{\sqrt{{\gamma^{yy}}}}}}$
for eigenvalue ${\lambda} = {{-{{\beta^y}}}{-{{{\alpha}} \cdot {{\sqrt{{\gamma^{yy}}}}}}}}$
eigenvectors are
$\left[\begin{array}{c} \sqrt{{\gamma^{yy}}}\\ 0\\ 1\\ 0\end{array}\right]$
for eigenvalue ${\lambda} = {-{{\beta^y}}}$
eigenvectors are
$\left[\begin{array}{cc} 0& 0\\ -{{\frac{1}{{\gamma^{yx}}}} {{\gamma^{yy}}}}& -{{\frac{1}{{\gamma^{yx}}}} {{\gamma^{yz}}}}\\ 1& 0\\ 0& 1\end{array}\right]$
for eigenvalue ${\lambda} = {{-{{\beta^y}}} + {{{\alpha}} \cdot {{\sqrt{{\gamma^{yy}}}}}}}$
eigenvectors are
$\left[\begin{array}{c} -{\sqrt{{\gamma^{yy}}}}\\ 0\\ 1\\ 0\end{array}\right]$
multiplicities 1, 2, 1
$\Lambda$:
$\left[\begin{array}{cccc} {-{{\beta^y}}}{-{{{\alpha}} \cdot {{\sqrt{{\gamma^{yy}}}}}}}& 0& 0& 0\\ 0& -{{\beta^y}}& 0& 0\\ 0& 0& -{{\beta^y}}& 0\\ 0& 0& 0& {-{{\beta^y}}} + {{{\alpha}} \cdot {{\sqrt{{\gamma^{yy}}}}}}\end{array}\right]$
R:
$\left[\begin{array}{cccc} \sqrt{{\gamma^{yy}}}& 0& 0& -{\sqrt{{\gamma^{yy}}}}\\ 0& -{{\gamma^{yy}}}& -{{\gamma^{yz}}}& 0\\ 1& {\gamma^{yx}}& 0& 1\\ 0& 0& {\gamma^{yx}}& 0\end{array}\right]$
L:
$\left[\begin{array}{cccc} \frac{1}{{{2}} {{\sqrt{{\gamma^{yy}}}}}}& \frac{{\gamma^{yx}}}{{{2}} {{{\gamma^{yy}}}}}& \frac{1}{2}& \frac{{\gamma^{yz}}}{{{2}} {{{\gamma^{yy}}}}}\\ 0& -{\frac{1}{{\gamma^{yy}}}}& 0& -{\frac{{\gamma^{yz}}}{{{{\gamma^{yx}}}} \cdot {{{\gamma^{yy}}}}}}\\ 0& 0& 0& \frac{1}{{\gamma^{yx}}}\\ -{\frac{1}{{{2}} {{\sqrt{{\gamma^{yy}}}}}}}& \frac{{\gamma^{yx}}}{{{2}} {{{\gamma^{yy}}}}}& \frac{1}{2}& \frac{{\gamma^{yz}}}{{{2}} {{{\gamma^{yy}}}}}\end{array}\right]$
A check: $\left[\begin{array}{cccc} -{{\beta^y}}& -{{{\alpha}} \cdot {{{\gamma^{yx}}}}}& -{{{\alpha}} \cdot {{{\gamma^{yy}}}}}& -{{{\alpha}} \cdot {{{\gamma^{yz}}}}}\\ 0& -{{\beta^y}}& 0& 0\\ -{\alpha}& 0& -{{\beta^y}}& 0\\ 0& 0& 0& -{{\beta^y}}\end{array}\right]$ A diff: $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

---

${{ A} _z} = {\left[\begin{array}{cccc} -{{\beta^z}}& {-{\alpha}} {{{\gamma^{zx}}}}& {-{\alpha}} {{{\gamma^{zy}}}}& {-{\alpha}} {{{\gamma^{zz}}}}\\ 0& -{{\beta^z}}& 0& 0\\ 0& 0& -{{\beta^z}}& 0\\ -{\alpha}& 0& 0& -{{\beta^z}}\end{array}\right]}$
char poly:
${{{{\left({{{\beta^z}} + {\lambda}}\right)}^{2}}} {{\left({{{\left({{{\beta^z}} + {\lambda}}\right)}^{2}}{-{{{{\gamma^{zz}}}} \cdot {{{\alpha}^{2}}}}}}\right)}}} = {0}$
lambdas ${-{{\beta^z}}}{-{{{\alpha}} \cdot {{\sqrt{{\gamma^{zz}}}}}}}$, $-{{\beta^z}}$, ${-{{\beta^z}}} + {{{\alpha}} \cdot {{\sqrt{{\gamma^{zz}}}}}}$
for eigenvalue ${\lambda} = {{-{{\beta^z}}}{-{{{\alpha}} \cdot {{\sqrt{{\gamma^{zz}}}}}}}}$
eigenvectors are
$\left[\begin{array}{c} \sqrt{{\gamma^{zz}}}\\ 0\\ 0\\ 1\end{array}\right]$
for eigenvalue ${\lambda} = {-{{\beta^z}}}$
eigenvectors are
$\left[\begin{array}{cc} 0& 0\\ -{{\frac{1}{{\gamma^{zx}}}} {{\gamma^{zy}}}}& -{{\frac{1}{{\gamma^{zx}}}} {{\gamma^{zz}}}}\\ 1& 0\\ 0& 1\end{array}\right]$
for eigenvalue ${\lambda} = {{-{{\beta^z}}} + {{{\alpha}} \cdot {{\sqrt{{\gamma^{zz}}}}}}}$
eigenvectors are
$\left[\begin{array}{c} -{\sqrt{{\gamma^{zz}}}}\\ 0\\ 0\\ 1\end{array}\right]$
multiplicities 1, 2, 1
$\Lambda$:
$\left[\begin{array}{cccc} {-{{\beta^z}}}{-{{{\alpha}} \cdot {{\sqrt{{\gamma^{zz}}}}}}}& 0& 0& 0\\ 0& -{{\beta^z}}& 0& 0\\ 0& 0& -{{\beta^z}}& 0\\ 0& 0& 0& {-{{\beta^z}}} + {{{\alpha}} \cdot {{\sqrt{{\gamma^{zz}}}}}}\end{array}\right]$
R:
$\left[\begin{array}{cccc} \sqrt{{\gamma^{zz}}}& 0& 0& -{\sqrt{{\gamma^{zz}}}}\\ 0& -{{\gamma^{zy}}}& -{{\gamma^{zz}}}& 0\\ 0& {\gamma^{zx}}& 0& 0\\ 1& 0& {\gamma^{zx}}& 1\end{array}\right]$
L:
$\left[\begin{array}{cccc} \frac{1}{{{2}} {{\sqrt{{\gamma^{zz}}}}}}& \frac{{\gamma^{zx}}}{{{2}} {{{\gamma^{zz}}}}}& \frac{{\gamma^{zy}}}{{{2}} {{{\gamma^{zz}}}}}& \frac{1}{2}\\ 0& 0& \frac{1}{{\gamma^{zx}}}& 0\\ 0& -{\frac{1}{{\gamma^{zz}}}}& -{\frac{{\gamma^{zy}}}{{{{\gamma^{zx}}}} \cdot {{{\gamma^{zz}}}}}}& 0\\ -{\frac{1}{{{2}} {{\sqrt{{\gamma^{zz}}}}}}}& \frac{{\gamma^{zx}}}{{{2}} {{{\gamma^{zz}}}}}& \frac{{\gamma^{zy}}}{{{2}} {{{\gamma^{zz}}}}}& \frac{1}{2}\end{array}\right]$
A check: $\left[\begin{array}{cccc} -{{\beta^z}}& -{{{\alpha}} \cdot {{{\gamma^{zx}}}}}& -{{{\alpha}} \cdot {{{\gamma^{zy}}}}}& -{{{\alpha}} \cdot {{{\gamma^{zz}}}}}\\ 0& -{{\beta^z}}& 0& 0\\ 0& 0& -{{\beta^z}}& 0\\ -{\alpha}& 0& 0& -{{\beta^z}}\end{array}\right]$ A diff: $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$