Editing QR decomposition (section)

===Square matrix===
Any real [[square matrix]] ''A'' may be decomposed as
: <math> A = QR, </math>

where ''Q'' is an [[orthogonal matrix]] (its columns are [[orthogonal]] [[unit vector]]s meaning {{nowrap|<math>Q^\textsf{T} = Q^{-1}</math>)}} and ''R'' is an upper [[triangular matrix]] (also called right triangular matrix). If ''A'' is [[invertible matrix|invertible]], then the factorization is unique if we require the diagonal elements of ''R'' to be positive.

If instead ''A'' is a complex square matrix, then there is a decomposition ''A'' = ''QR'' where ''Q'' is a [[unitary matrix]] (so the [[conjugate transpose]] {{nowrap|<math>Q^\dagger = Q^{-1}</math>).}}

If ''A'' has ''n'' [[linearly independent]] columns, then the first ''n'' columns of ''Q'' form an [[orthonormal basis]] for the [[column space]] of ''A''. More generally, the first ''k'' columns of ''Q'' form an orthonormal basis for the [[linear span|span]] of the first ''k'' columns of ''A'' for any {{nowrap|1 ≤ ''k'' ≤ ''n''}}.<ref name="Trefethen">{{cite book |last1=Trefethen |first1=Lloyd N. |last2=Bau |first2=David III |author1-link=Nick Trefethen |title=Numerical linear algebra |date=1997 |publisher=[[Society for Industrial and Applied Mathematics]] |location=Philadelphia, PA |isbn=978-0-898713-61-9}}</ref>  The fact that any column ''k'' of ''A'' only depends on the first ''k'' columns of ''Q'' corresponds to the triangular form of&nbsp;''R''.<ref name=Trefethen/>