Utilize o algoritmo de fatoração L D L^{\prime} para descobrir uma fatoração da forma A=L D L^{\prime} para as seguintes matrizes:
A=\left[\begin{array}{rrrr}4 & 1 & 1 & 1 \\\\ 1 & 3 & 0 & -1 \\\\ 1 & 0 & 2 & 1 \\\\ 1 & -1 & 1 & 4\end{array}\right]

Passo 1

O enunciado pede a fatoração LDL′ de
A=\begin{pmatrix} 4 & 1 & 1 & 1\\\\ 1 & 3 & 0 & -1\\\\ 1 & 0 & 2 & 1\\\\ 1 & -1 & 1 & 4 \end{pmatrix},
isto é, encontrar matriz unitária inferior L e diagonal D tais que A=L\,D\,L^{\prime}.
Usamos o algoritmo recursivo: para k=1,\dots,n,
d_{k}=a_{kk}-\sum_{j=1}^{k-1}l_{kj}^{2}d_{j}, \quad l_{ik}=\frac{a_{ik}-\sum_{j=1}^{k-1}l_{ij}l_{kj}d_{j}}{d_{k}} \quad(i>k),
com l_{kk}=1.

Passo 2

Para k=1:
d_{1}=a_{11}=4.
Para i=2,3,4:
l_{i1}=\frac{a_{i1}}{d_{1}} \;\Longrightarrow\; l_{21}=\tfrac{1}{4},\; l_{31}=\tfrac{1}{4},\; l_{41}=\tfrac{1}{4}.

Passo 3

Para k=2:
d_{2}=a_{22}-l_{21}^{2}d_{1} =3-\bigl(\tfrac14\bigr)^{2}\cdot4 =\tfrac{11}{4}.
Para i=3,4:
l_{i2} =\frac{a_{i2}-l_{i1}l_{21}d_{1}}{d_{2}} \;\Longrightarrow\; l_{32}=\frac{0-(\tfrac14\cdot\tfrac14\cdot4)}{\tfrac{11}{4}}=-\tfrac{1}{11}, \; l_{42}=\frac{-1-(\tfrac14\cdot\tfrac14\cdot4)}{\tfrac{11}{4}}=-\tfrac{5}{11}.

Passo 4

Para k=3:
d_{3} =a_{33}-\bigl(l_{31}^{2}d_{1}+l_{32}^{2}d_{2}\bigr) =2-\Bigl(\tfrac1{4}^{2}\cdot4+(-\tfrac1{11})^{2}\cdot\tfrac{11}{4}\Bigr) =\frac{209}{121}.
Para i=4:
l_{43} =\frac{a_{43}-\bigl(l_{41}l_{31}d_{1}+l_{42}l_{32}d_{2}\bigr)}{d_{3}} =\frac{1-\bigl(\tfrac14\cdot\tfrac14\cdot4+(-\tfrac5{11})(-\tfrac1{11})\tfrac{11}{4}\bigr)}{\tfrac{209}{121}} =\frac{7}{19}.

Passo 5

Para k=4:
d_{4} =a_{44}-\bigl(l_{41}^{2}d_{1}+l_{42}^{2}d_{2}+l_{43}^{2}d_{3}\bigr) =4-\Bigl(\tfrac1{4}+\tfrac{25}{121}\cdot\tfrac{11}{4}+\tfrac{49}{361}\cdot\tfrac{209}{121}\Bigr) =\frac{56}{19}.

Resposta

Os fatores ficam
L=\begin{pmatrix} 1 & 0 & 0 & 0\\\ \frac{1}{4} & 1 & 0 & 0\\\ \frac{1}{4} & -\frac{1}{11} & 1 & 0\\\ \frac{1}{4} & -\frac{5}{11} & \frac{7}{19} & 1 \end{pmatrix}, \quad D=\begin{pmatrix} 4 & 0 & 0 & 0\\\\ 0 & \tfrac{11}{4} & 0 & 0\\\\ 0 & 0 & \tfrac{209}{121} & 0\\\\ 0 & 0 & 0 & \tfrac{56}{19} \end{pmatrix},
de modo que A=L\,D\,L^{\prime}.

Exercícios de Livros Relacionados