シュール分解
↑ Horn & Johnson 1985, Section 2.3 and further (p. 79)
↑ Golub & van Loan 1996, Section 7.7 (Schur Decomposition)
↑ Golub & van Loan 1996, section 7.7 (QZ)
↑ Daniel Kressner: "Numerical Methods for General and Structured Eigenvalue Problems", Chap2, Springer, LNCSE-46 (2005).
= [ 1 0 0 0 0 Q 2 ~ − 1 ] Q 1 − 1 A Q 1 [ 1 0 0 0 0 Q 2 ~ ] = [ − 5 − 3 / 2 35 / 6 0 0 Q 2 ~ − 1 A ~ Q 2 ~ ] = [ − 5 − 3 / 2 35 / 6 0 − 5 8 / 3 0 0 3 ] {\displaystyle {\begin{aligned}&={\begin{bmatrix}1&{\begin{matrix}0&0\end{matrix}}\\{\begin{matrix}0\\0\end{matrix}}&{\tilde {Q_{2}}}^{-1}\end{bmatrix}}{Q_{1}}^{-1}AQ_{1}{\begin{bmatrix}1&{\begin{matrix}0&0\end{matrix}}\\{\begin{matrix}0\\0\end{matrix}}&{\tilde {Q_{2}}}\end{bmatrix}}\\&={\begin{bmatrix}-5&{\begin{matrix}-3/{\sqrt {2}}&35/{\sqrt {6}}\end{matrix}}\\{\begin{matrix}0\\0\end{matrix}}&{\tilde {Q_{2}}}^{-1}{\tilde {A}}{\tilde {Q_{2}}}\end{bmatrix}}={\begin{bmatrix}-5&-3/{\sqrt {2}}&35/{\sqrt {6}}\\0&-5&8/{\sqrt {3}}\\0&0&3\end{bmatrix}}\end{aligned}}}
最後の上三角化行列を U とすれば、ユニタリ行列 Q := Q 1 Q 2 {\displaystyle Q:=Q_{1}Q_{2}} により Q − 1 A Q = U , A = Q U Q − 1 {\displaystyle Q^{-1}AQ=U,A=QUQ^{-1}}