A I J = ∂ E ∂ x n x + ∂ F ∂ y n y + ∂ G ∂ z n z {\displaystyle \mathbf {A} _{IJ}={\frac {\partial E}{\partial x}}n_{x}+{\frac {\partial F}{\partial y}}n_{y}+{\frac {\partial G}{\partial z}}n_{z}}
Diagonalizing:
corresponding to eigenvalue λ 1 = | V → + a | {\displaystyle \lambda _{1}=|{\vec {V}}+a|}
corresponding to eigenvalue λ 2 = | V → − a | {\displaystyle \lambda _{2}=|{\vec {V}}-a|}
corresponding to eigenvalue λ 3 = | V → | {\displaystyle \lambda _{3}=|{\vec {V}}|}
P − 1 = [ γ ^ V → 2 2 + V → a − a n x − γ ^ u − a n y − γ ^ v − a n z − γ ^ w γ ^ γ ^ V → 2 2 − V → a a n x − γ ^ u a n y − γ ^ v a n z − γ ^ w γ ^ 2 a ( w n y − v n z + a n x ) − n x V → 2 γ ^ 2 u n x γ ^ 2 v n x γ ^ + 2 a n z 2 w n x γ ^ − 2 a n y − 2 n x γ ^ 2 a ( u n z − w n x + a n y ) − n y V → 2 γ ^ 2 u n y γ ^ + 2 a n z 2 v n y γ ^ 2 w n y γ ^ + 2 a n x − 2 n y γ ^ 2 a ( v n x − u n y + a n z ) − n z V → 2 γ ^ 2 u n z γ ^ + 2 a n y 2 v n z γ ^ − 2 a n x 2 w n z γ ^ − 2 n z γ ^ ] 1 2 a 2 {\displaystyle P^{-1}=\left[{\begin{array}{c c c c c}{\hat {\gamma }}{\dfrac {{\vec {V}}^{2}}{2}}+{\vec {V}}a&-an_{x}-{\hat {\gamma }}u&-an_{y}-{\hat {\gamma }}v&-an_{z}-{\hat {\gamma }}w&{\hat {\gamma }}\\\\{\hat {\gamma }}{\dfrac {{\vec {V}}^{2}}{2}}-{\vec {V}}a&an_{x}-{\hat {\gamma }}u&an_{y}-{\hat {\gamma }}v&an_{z}-{\hat {\gamma }}w&{\hat {\gamma }}\\\\2a(wn_{y}-vn_{z}+an_{x})-n_{x}{\vec {V}}^{2}{\hat {\gamma }}&2un_{x}{\hat {\gamma }}&2vn_{x}{\hat {\gamma }}+2an_{z}&2wn_{x}{\hat {\gamma }}-2an_{y}&-2n_{x}{\hat {\gamma }}\\\\2a(un_{z}-wn_{x}+an_{y})-n_{y}{\vec {V}}^{2}{\hat {\gamma }}&2un_{y}{\hat {\gamma }}+2an_{z}&2vn_{y}{\hat {\gamma }}&2wn_{y}{\hat {\gamma }}+2an_{x}&-2n_{y}{\hat {\gamma }}\\\\2a(vn_{x}-un_{y}+an_{z})-n_{z}{\vec {V}}^{2}{\hat {\gamma }}&2un_{z}{\hat {\gamma }}+2an_{y}&2vn_{z}{\hat {\gamma }}-2an_{x}&2wn_{z}{\hat {\gamma }}&-2n_{z}{\hat {\gamma }}\\\end{array}}\right]{\frac {1}{2a^{2}}}}
![{\displaystyle A_{x}=\left[{\begin{array}{c c c c c}0&1&0&0&0\\{\hat {\gamma }}H-u^{2}-a^{2}&(3-\gamma )u&-v{\hat {\gamma }}&-w{\hat {\gamma }}&{\hat {\gamma }}\\-uv&v&u&0&0\\-uw&w&0&u&0\\u[(\gamma -2)H-a^{2}]&H-{\hat {\gamma }}u^{2}&-uv{\hat {\gamma }}&-uw{\hat {\gamma }}&\gamma u\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abc2c304f3a059fc8b5f7af53bd6f1d444c120f2)
![{\displaystyle P_{x}=\left[{\begin{array}{c c c c c}1&1&0&0&1\\u-a&u&0&0&u+a\\v&v&1&0&v\\w&w&0&1&w\\H-ua&{\frac {1}{2}}{\vec {V}}^{2}&v&w&H+ua\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2841a4072c650fc649b83f73dce0727948ec7d1f)
![{\displaystyle \mathbf {\Lambda } _{x}=\left[{\begin{array}{c c c c c}u-a&&&&\\&u&&&\\&&u&&\\&&&u&\\&&&&u+a\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5f51e7bf8e4bacd7157acfd4396e523d7ba3df)
![{\displaystyle P_{x}^{-1}=\left[{\begin{array}{c c c c c}{\dfrac {1}{2}}{\vec {V}}^{2}+{\dfrac {ua}{\gamma -1}}&-u-{\dfrac {a}{\gamma -1}}&-v&-w&1\\\\2\left({\dfrac {a^{2}}{\gamma -1}}-{\dfrac {{\vec {V}}^{2}}{2}}\right)&2u&2v&2w&-2\\\\-v{\dfrac {2a^{2}}{\gamma -1}}&0&{\dfrac {2a^{2}}{\gamma -1}}&0&0\\\\-w{\dfrac {2a^{2}}{\gamma -1}}&0&0&{\dfrac {2a^{2}}{\gamma -1}}&0\\\\{\dfrac {1}{2}}{\vec {V}}^{2}-{\dfrac {ua}{\gamma -1}}&{\dfrac {a}{\gamma -1}}-u&-v&-w&1\end{array}}\right]{\frac {\gamma -1}{2a^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/213baa1a52c0542985f122a58da57a357dd1e4fa)
![{\displaystyle P_{y}=\left[{\begin{array}{c c c c c}1&1&0&0&1\\u&u&1&0&u\\v-a&v&0&0&v+a\\w&w&0&1&w\\H-va&{\frac {1}{2}}{\vec {V}}^{2}&u&w&H+va\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d1ccd3164f7c1661e800ab4ab4dcf55873ddd)
![{\displaystyle A_{IJ}=\left[{\begin{array}{c c c c c}0&n_{x}&n_{y}&n_{z}&0\\\\\left[{\hat {\gamma }}H-a^{2}\right]n_{x}-u{\vec {V}}&(2-\gamma )un_{x}+{\vec {V}}&un_{y}-v{\hat {\gamma }}n_{x}&un_{z}-w{\hat {\gamma }}n_{x}&{\hat {\gamma }}n_{x}\\\\\left[{\hat {\gamma }}H-a^{2}\right]n_{y}-v{\vec {V}}&vn_{x}-u{\hat {\gamma }}n_{y}&(2-\gamma )vn_{y}+{\vec {V}}&vn_{z}-w{\hat {\gamma }}n_{y}&{\hat {\gamma }}n_{y}\\\\\left[{\hat {\gamma }}H-a^{2}\right]n_{z}-w{\vec {V}}&wn_{x}-u{\hat {\gamma }}n_{z}&wn_{y}-v{\hat {\gamma }}n_{z}&(2-\gamma )wn_{z}+{\vec {V}}&{\hat {\gamma }}n_{z}\\\\{\vec {V}}\left[(\gamma -2)H-a^{2}\right]&Hn_{x}-u{\vec {V}}{\hat {\gamma }}&Hn_{y}-v{\vec {V}}{\hat {\gamma }}&Hn_{z}-w{\vec {V}}{\hat {\gamma }}&\gamma {\vec {V}}\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af084e371f5e47b2ee54bcf206e0efe553d296ce)
![{\displaystyle R_{1}=\left[{\begin{array}{c}1\\u+an_{x}\\v+an_{y}\\w+an_{z}\\H+{\vec {V}}a{\vec {n}}\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7051a6fe91195898ce05974a61c7cfae1c3d4155)
![{\displaystyle R_{2}=\left[{\begin{array}{c}1\\u-an_{x}\\v-an_{y}\\w-an_{z}\\H-{\vec {V}}a{\vec {n}}\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d053cb0503d2d5ab021dc74c7833dda2038eb92b)
![{\displaystyle R_{3}=\left[{\begin{array}{c}n_{x}\\un_{x}\\vn_{x}+an_{z}\\wn_{x}-an_{y}\\{\dfrac {{\vec {V}}^{2}}{2}}n_{x}+a(vn_{z}-wn_{y})\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca4cc68296df397575800cafc5f65c4e5b341a6)
![{\displaystyle R_{4}=\left[{\begin{array}{c}n_{y}\\un_{y}-an_{z}\\vn_{y}\\wn_{y}+an_{x}\\{\dfrac {{\vec {V}}^{2}}{2}}n_{y}+a(wn_{x}-un_{z})\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecd81edc4c4de796c42da81ac5a92cd3f942898a)
![{\displaystyle R_{5}=\left[{\begin{array}{c}n_{z}\\un_{z}+an_{y}\\vn_{z}-an_{x}\\wn_{z}\\{\dfrac {{\vec {V}}^{2}}{2}}n_{z}+a(un_{y}-vn_{x})\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44ce24373b51dfd9ea1c344feceb5ee2aec4808a)
![{\displaystyle P^{-1}=\left[{\begin{array}{c c c c c}{\hat {\gamma }}{\dfrac {{\vec {V}}^{2}}{2}}+{\vec {V}}a&-an_{x}-{\hat {\gamma }}u&-an_{y}-{\hat {\gamma }}v&-an_{z}-{\hat {\gamma }}w&{\hat {\gamma }}\\\\{\hat {\gamma }}{\dfrac {{\vec {V}}^{2}}{2}}-{\vec {V}}a&an_{x}-{\hat {\gamma }}u&an_{y}-{\hat {\gamma }}v&an_{z}-{\hat {\gamma }}w&{\hat {\gamma }}\\\\2a(wn_{y}-vn_{z}+an_{x})-n_{x}{\vec {V}}^{2}{\hat {\gamma }}&2un_{x}{\hat {\gamma }}&2vn_{x}{\hat {\gamma }}+2an_{z}&2wn_{x}{\hat {\gamma }}-2an_{y}&-2n_{x}{\hat {\gamma }}\\\\2a(un_{z}-wn_{x}+an_{y})-n_{y}{\vec {V}}^{2}{\hat {\gamma }}&2un_{y}{\hat {\gamma }}+2an_{z}&2vn_{y}{\hat {\gamma }}&2wn_{y}{\hat {\gamma }}+2an_{x}&-2n_{y}{\hat {\gamma }}\\\\2a(vn_{x}-un_{y}+an_{z})-n_{z}{\vec {V}}^{2}{\hat {\gamma }}&2un_{z}{\hat {\gamma }}+2an_{y}&2vn_{z}{\hat {\gamma }}-2an_{x}&2wn_{z}{\hat {\gamma }}&-2n_{z}{\hat {\gamma }}\\\end{array}}\right]{\frac {1}{2a^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/900756bac1f9ff1fffce1afb5842ceed6adf681c)