Schur decomposition, proof of

The columns of the unitary matrix $Q$ in Schur’s decomposition theorem form an orthonormal basis of $\\mathbb{C}^{n}$ . The matrix $A$ takes the upper-triangular form $D+N$ on this basis. Conversely, if $v_{1},\\ldots,v_{n}$ is an orthonormal basis for which $A$ is of this form then the matrix $Q$ with $v_{i}$ as its $i$ -th column satisfies the theorem.

To find such a basis we proceed by induction on $n$ . For $n=1$ we can simply take $Q=1$ . If $n>1$ then let $v\\in\\mathbb{C}^{n}$ be an eigenvector of $A$ of unit length and let $V=v^{\\perp}$ be its orthogonal complement. If $\\pi$ denotes the orthogonal projection onto the line spanned by $v$ then $(1-\\pi)A$ maps $V$ into $V$ .

By induction there is an orthonormal basis $v_{2},\\ldots,v_{n}$ of $V$ for which $(1-\\pi)A$ takes the desired form on $V$ . Now $A=\\pi A+(1-\\pi)A$ so $Av_{i}\\equiv(1-\\pi)Av_{i}(\\mod v)$ for $i\\in\\{2,\\ldots,n\\}$ . Then $v,v_{2},\\ldots,v_{n}$ can be used as a basis for the Schur decomposition on $\\mathbb{C}^{n}$ .

单词	SchurDecompositionProofOf
释义	Schur decomposition, proof of The columns of the unitary matrix $Q$ in Schur’s decomposition theorem form an orthonormal basis of $\\mathbb{C}^{n}$ . The matrix $A$ takes the upper-triangular form $D+N$ on this basis. Conversely, if $v_{1},\\ldots,v_{n}$ is an orthonormal basis for which $A$ is of this form then the matrix $Q$ with $v_{i}$ as its $i$ -th column satisfies the theorem. To find such a basis we proceed by induction on $n$ . For $n=1$ we can simply take $Q=1$ . If $n>1$ then let $v\\in\\mathbb{C}^{n}$ be an eigenvector of $A$ of unit length and let $V=v^{\\perp}$ be its orthogonal complement. If $\\pi$ denotes the orthogonal projection onto the line spanned by $v$ then $(1-\\pi)A$ maps $V$ into $V$ . By induction there is an orthonormal basis $v_{2},\\ldots,v_{n}$ of $V$ for which $(1-\\pi)A$ takes the desired form on $V$ . Now $A=\\pi A+(1-\\pi)A$ so $Av_{i}\\equiv(1-\\pi)Av_{i}(\\mod v)$ for $i\\in\\{2,\\ldots,n\\}$ . Then $v,v_{2},\\ldots,v_{n}$ can be used as a basis for the Schur decomposition on $\\mathbb{C}^{n}$ .
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