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In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. == Statement == The Schur decomposition reads as follows: if ''A'' is a ''n'' × ''n'' square matrix with complex entries, then ''A'' can be expressed as〔(Section 2.3 and further at (p. 79 ))〕〔(Section 7.7 at (p. 313 ))〕 : where ''Q'' is a unitary matrix (so that its inverse ''Q''−1 is also the conjugate transpose ''Q'' * of ''Q''), and ''U'' is an upper triangular matrix, which is called a Schur form of ''A''. Since ''U'' is similar to ''A'', it has the same multiset of eigenvalues, and since it is triangular, those eigenvalues are the diagonal entries of ''U''. The Schur decomposition implies that there exists a nested sequence of ''A''-invariant subspaces = ''V''0 ⊂ ''V''1 ⊂ ... ⊂ ''Vn'' = C''n'', and that there exists an ordered orthonormal basis (for the standard Hermitian form of C''n'') such that the first ''i'' basis vectors span ''V''''i'' for each ''i'' occurring in the nested sequence. Phrased somewhat differently, the first part says that a linear operator ''J'' on a complex finite-dimensional vector space stabilizes a complete flag (''V''1,...,''Vn''). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Schur decomposition」の詳細全文を読む スポンサード リンク
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