Editing QR decomposition (section)

===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).