3.8: Factorización QR
- Page ID
- 119038
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)El proceso Gram-Schmidt conduce naturalmente a una factorización matricial. Dejar\(\text{A}\) ser una\(n\) matriz\(m\) -by- con columnas\(n\) linealmente independientes dadas por\(\{\text{x}_1,\: \text{x}_2,\cdots , \text{x}_n\}\). Siguiendo el proceso de Gram-Schmidt, siempre es posible construir una base ortornormal para el espacio de columna de\(\text{A}\), denotado por\(\{\text{q}_1,\: \text{q}_2,\cdots , \text{q}_n\}\). Una característica importante de esta base ortonormal es que los primeros vectores de\(k\) base del conjunto ortonormal abarcan el mismo subespacio vectorial que las primeras\(k\) columnas de la matriz\(\text{A}\). Para algunos coeficientes\(r_{ij}\), por lo tanto, podemos escribir
\[\begin{aligned}\text{x}_1&=r_{11}\text{q}_1, \\ \text{x}_2&=r_{12}\text{q}_1+r_{22}\text{q}_2, \\ \text{x}_3&=r_{13}\text{q}_1+r_{23}\text{q}_2+r_{33}\text{q}_3, \\ \vdots&\qquad \vdots \\ \text{x}_n&=r_{1\text{n}}\text{q}_1+r_{2\text{n}}\text{q}_2+\cdots +r_{\text{nn}}\text{q}_{\text{n}};\end{aligned} \nonumber \]
y estas ecuaciones se pueden escribir en forma de matriz como
\[\left(\begin{array}{ccccc}|&|&|&&| \\ \text{x}_1&\text{x}_2&\text{x}_3&\cdots&\text{x}_{\text{n}} \\ |&|&|&&|\end{array}\right)=\left(\begin{array}{ccccc}|&|&|&&| \\ \text{q}_1&\text{q}_2&\text{q}_3&\cdots&\text{q}_{\text{n}} \\ |&|&|&&|\end{array}\right)\left(\begin{array}{cccc}r_{11}&r_{12}&\cdots &r_{1n} \\ 0&r_{22}&\cdots &r_{2n} \\ \vdots&\vdots&\vdots&\vdots \\ 0&0&\cdots &r_{nn}\end{array}\right).\nonumber \]
Esta forma representa la factorización matricial llamada factorización QR, y generalmente se escribe como
\[\text{A}=\text{QR},\nonumber \]
donde\(\text{Q}\) es una matriz ortogonal y\(\text{R}\) es una matriz triangular superior. Los elementos diagonales de también se\(\text{R}\) pueden hacer no negativos ajustando adecuadamente los signos de los vectores de base ortonormal.
Como ejemplo concreto, encontraremos la\(\text{QR}\) factorización de la matriz
\[\text{A}=\left(\begin{array}{cc}1&2\\2&1\end{array}\right)=\left(\begin{array}{cc}|&|\\ \text{a}_1&\text{a}_2 \\ |&|\end{array}\right).\nonumber \]
Aplicando el proceso Gram-Schmidt a los vectores de columna de\(\text{A}\), tenemos para los vectores ortogonales no normalizados
\[\begin{aligned}\text{q}_1&=\text{a}_1=\left(\begin{array}{c}1\\2\end{array}\right), \\ \text{q}_2&=\text{a}_2-\frac{(\text{q}_1^{\text{T}}\text{a}_2)\text{q}_1}{\text{q}_1^{\text{T}}\text{q}_1}=\left(\begin{array}{c}2\\1\end{array}\right)-\frac{4}{5}\left(\begin{array}{c}1\\2\end{array}\right)=\left(\begin{array}{r}6/5 \\ -3/5\end{array}\right)=\frac{3}{5}\left(\begin{array}{r}2\\-1\end{array}\right),\end{aligned} \nonumber \]
y normalizando, obtenemos
\[\text{q}_1=\frac{1}{\sqrt{5}}\left(\begin{array}{c}1\\2\end{array}\right),\quad\text{q}_2=\frac{1}{\sqrt{5}}\left(\begin{array}{r}2\\-1\end{array}\right).\nonumber \]
La proyección de las columnas de\(\text{A}\) sobre el conjunto de vectores ortonormales viene dada por
\[\text{a}_1=(\text{a}_1^{\text{T}}\text{q}_1)\text{q}_1,\quad\text{a}_2=(\text{a}_2^{\text{T}}\text{q}_1)\text{q}_1+(\text{a}_2^{\text{T}}\text{q}_2)\text{q}_2,\nonumber \]
y con\(r_{ij}=\text{a}_j^{\text{T}}\text{q}_i\), calculamos
\[\begin{aligned}r_{11}&=\text{a}_1^{\text{T}}\text{q}_1=\left(\begin{array}{cc}1&2\end{array}\right)\left(\begin{array}{c}1\\2\end{array}\right)\frac{1}{\sqrt{5}}=\sqrt{5}, \\ r_{12}&=\text{a}_2^{\text{T}}\text{q}_1=\left(\begin{array}{cc}2&1\end{array}\right)\left(\begin{array}{c}1\\2\end{array}\right)\frac{1}{\sqrt{5}}=\frac{4\sqrt{5}}{5}, \\ r_{22}&=\text{a}_2^{\text{T}}\text{q}_2=\left(\begin{array}{cc}2&1\end{array}\right)\left(\begin{array}{c}2\\-1\end{array}\right)\frac{1}{\sqrt{5}}=\frac{3\sqrt{5}}{5}.\end{aligned} \nonumber \]
Por lo tanto,\(\text{A}\) se da la\(\text{QR}\) factorización de
\[\left(\begin{array}{cc}1&2\\2&1\end{array}\right)=\left(\begin{array}{cc}1/\sqrt{5}&2/\sqrt{5} \\ 2/\sqrt{5}&-1/\sqrt{5}\end{array}\right)\left(\begin{array}{cc}\sqrt{5}&4\sqrt{5}/5 \\ 0&3\sqrt{5}/5\end{array}\right).\nonumber \]