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Student[LinearAlgebra]

 QRDecomposition
 compute the QR factorization of a Matrix

 Calling Sequence QRDecomposition(A, options)

Parameters

 A - Matrix options - (optional) parameters; for a complete list, see LinearAlgebra[QRDecomposition]

Description

 • The QRDecomposition command computes the QR decomposition of the Matrix A, that is, a factorization into a product of an orthogonal (or unitary) Matrix, Q, and an upper triangular Matrix, R, such that $A=Q\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}R$.  The Matrices Q and R are returned in an expression sequence.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):$
 > $A≔⟨⟨1,2,3⟩|⟨4,5,6⟩⟩$
 ${A}{:=}\left[\begin{array}{rr}{1}& {4}\\ {2}& {5}\\ {3}& {6}\end{array}\right]$ (1)
 > $Q,R≔\mathrm{QRDecomposition}\left(A\right)$
 ${Q}{,}{R}{:=}\left[\begin{array}{ccc}\frac{{1}}{{14}}{}\sqrt{{14}}& \frac{{4}}{{21}}{}\sqrt{{21}}& \frac{{1}}{{6}}{}\sqrt{{6}}\\ \frac{{1}}{{7}}{}\sqrt{{14}}& \frac{{1}}{{21}}{}\sqrt{{21}}& {-}\frac{{1}}{{3}}{}\sqrt{{6}}\\ \frac{{3}}{{14}}{}\sqrt{{14}}& {-}\frac{{2}}{{21}}{}\sqrt{{21}}& \frac{{1}}{{6}}{}\sqrt{{6}}\end{array}\right]{,}\left[\begin{array}{cc}\sqrt{{14}}& \frac{{16}}{{7}}{}\sqrt{{14}}\\ {0}& \frac{{3}}{{7}}{}\sqrt{{21}}\\ {0}& {0}\end{array}\right]$ (2)
 > $\mathrm{Equal}\left(Q\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}R,A\right)$
 ${\mathrm{true}}$ (3)

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