Editing QR decomposition (section)

==Cases and definitions==

===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/>

===Rectangular matrix===
More generally, we can factor a complex ''m''×''n'' matrix ''A'', with {{nowrap|''m'' ≥ ''n''}}, as the product of an ''m''×''m'' [[unitary matrix]] ''Q'' and an ''m''×''n'' upper triangular matrix ''R''.  As the bottom (''m''−''n'') rows of an ''m''×''n'' upper triangular matrix consist entirely of zeroes, it is often useful to partition ''R'', or both ''R'' and ''Q'':
:<math>
  A = QR = Q \begin{bmatrix} R_1 \\ 0 \end{bmatrix}
  = \begin{bmatrix} Q_1 & Q_2 \end{bmatrix} \begin{bmatrix} R_1 \\ 0 \end{bmatrix}
  = Q_1 R_1,
</math>

where ''R''<sub>1</sub> is an ''n''×''n'' upper triangular matrix, 0 is an {{nowrap|(''m'' − ''n'')×''n''}} [[zero matrix]], ''Q''<sub>1</sub> is ''m''×''n'', ''Q''<sub>2</sub> is {{nowrap|''m''×(''m'' − ''n'')}}, and ''Q''<sub>1</sub> and ''Q''<sub>2</sub> both have orthogonal columns.

{{harvtxt|Golub|Van Loan|1996|loc=§5.2}} call ''Q''<sub>1</sub>''R''<sub>1</sub> the ''thin QR factorization'' of ''A''; Trefethen and Bau call this the ''reduced QR factorization''.<ref name=Trefethen/> If ''A'' is of full [[matrix rank|rank]] ''n'' and we require that the diagonal elements of ''R''<sub>1</sub> are positive then ''R''<sub>1</sub> and ''Q''<sub>1</sub> are unique, but in general ''Q''<sub>2</sub> is not. ''R''<sub>1</sub> is then equal to the upper triangular factor of the [[Cholesky decomposition]] of ''A''{{starred}} ''A'' (=&nbsp;''A''<sup>T</sup>''A'' if ''A'' is real).

===QL, RQ and LQ decompositions===
Analogously, we can define QL, RQ, and LQ decompositions, with ''L'' being a ''lower'' triangular matrix.