Editing Schur decomposition (section)

== Statement ==
The complex Schur decomposition reads as follows: if {{mvar|A}} is an {{math|''n'' × ''n''}} [[square matrix]] with [[complex numbers|complex]] entries, then ''A'' can be expressed as<ref name=horn1985>{{cite book | last1 = Horn | first1 = R.A. | last2 = Johnson | first2 = C.R. | name-list-style=amp | year=1985 | title = Matrix Analysis | publisher = Cambridge University Press | isbn = 0-521-38632-2}} (Section 2.3 and further at [{{Google books|plainurl=y|id=PlYQN0ypTwEC|page=79|text=Schur}} p. 79])</ref><ref name=Golub1996>{{cite book|last1=Golub | first1= G.H. |last2=Van Loan | first2 = C.F. |name-list-style=amp |year=1996 |title=Matrix Computations | edition=3rd | publisher=Johns Hopkins University Press | isbn=0-8018-5414-8}}(Section 7.7 at [{{Google books|plainurl=y|id=mlOa7wPX6OYC|page=313|text=Schur Decomposition}} p. 313])</ref><ref>{{cite book |first=James R. |last=Schott |title=Matrix Analysis for Statistics |location=New York |publisher=John Wiley & Sons |year=2016 |edition=3rd |isbn=978-1-119-09247-6 |pages=175–178 |url=https://books.google.com/books?id=e-JFDAAAQBAJ&pg=PA177 }}</ref>
<math display="block"> A = Q U Q^{-1}</math>
for some [[unitary matrix]] ''Q'' (so that the inverse ''Q''<sup>−1</sup> is also the [[conjugate transpose]] ''Q''* of ''Q''), and some [[upper triangular matrix]] ''U''. This is called a '''Schur form''' of ''A''. Since ''U'' is [[similar (linear algebra)|similar]] to ''A'', it has the same [[Spectrum of a matrix|spectrum]], and since it is triangular, its [[eigenvalue]]s are the diagonal entries of ''U''.

The Schur decomposition implies that there exists a nested sequence of ''A''-invariant subspaces {{math|1={0} = ''V''<sub>0</sub> ⊂ ''V''<sub>1</sub> ⊂ ⋯ ⊂ ''V<sub>n</sub>'' = '''C'''<sup>''n''</sup>}}, and that there exists an ordered [[orthonormal basis]] (for the standard [[Hermitian form]] of {{math|'''C'''<sup>''n''</sup>}}) such that the first ''i'' basis vectors span {{math|''V''<sub>''i''</sub>}} 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 [[Orbit-stabilizer theorem#Orbits and stabilizers|stabilizes]] a complete [[Flag (linear algebra)|flag]] {{math|1=(''V''<sub>1</sub>, ..., ''V<sub>n</sub>'')}}.

There is also a real Schur decomposition. If {{mvar|A}} is an {{math|''n'' × ''n''}} [[square matrix]] with [[real numbers|real]] entries, then ''A'' can be expressed as<ref name=horn1985-2>{{cite book | last1 = Horn | first1 = R.A. | last2 = Johnson | first2 = C.R. | name-list-style=amp | year=1985 | title = Matrix Analysis | publisher = Cambridge University Press | isbn = 0-521-38632-2}} (Section 2.3 and further at [{{Google books|plainurl=y|id=PlYQN0ypTwEC|page=82|text=2.3.4}} p. 82])</ref> <math display="block"> A = Q H Q^{-1}</math> where {{mvar|Q}} is an [[orthogonal matrix]] and {{mvar|H}} is either upper or lower quasi-triangular. A quasi-triangular matrix is a matrix that when expressed as a block matrix of {{math|''2'' × ''2''}} and {{math|''1'' × ''1''}} blocks is triangular. This is a stronger property than being [[Hessenberg matrix|Hessenberg]]. Just as in the complex case, a family of commuting real matrices 
{''A<sub>i</sub>''} may be simultaneously brought to quasi-triangular form by an orthogonal matrix. There exists an orthogonal matrix ''Q'' such that, for every ''A<sub>i</sub>'' in the given family, 
<math display="block"> H_i = Q A_i Q^{-1}</math>
is upper quasi-triangular.