przykłady liczbowe do rozkładów macierzowych

Poniżej zamieszczone są przykłady liczbowe do ćwiczenia rozkładów, które można rozwiązać bez konieczności używania kalkulatora.

$A = LU$
$A = LDU’$
$A = LDL^{T}$
$A = QR$ (tylko st. dzienne)
$A = S\Lambda S^{-1}$
$A = Q\Lambda Q^{T}$ (tylko st. dzienne)

Zostały one wygenerowane automatycznie — jeżeli jest jakiś błąd to proszę o informację.

macierze 2x2

- Przykład 1 — $A_{2x2}$
  $$A = \left[ \begin{array}{rr} -4 & 7 \\ -3 & 9 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -4 & 7 \\ 0 & \frac{15}{4} \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -4 & 0 \\ 0 & \frac{15}{4} \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & -\frac{7}{4} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & -\frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 5 & -11 \\ 0 & 3 \\ \end{array}\right]$$
- Przykład 2 — $A_{2x2}$
  $$A = \left[ \begin{array}{rr} -4 & 9 \\ -3 & 8 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -4 & 9 \\ 0 & \frac{5}{4} \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -4 & 0 \\ 0 & \frac{5}{4} \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & -\frac{9}{4} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & -\frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 5 & -12 \\ 0 & 1 \\ \end{array}\right]$$
- Przykład 3 — $A_{2x2}$
  $$A = \left[ \begin{array}{rr} -8 & -9 \\ -6 & 7 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -8 & -9 \\ 0 & \frac{55}{4} \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -8 & 0 \\ 0 & \frac{55}{4} \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & \frac{9}{8} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & -\frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 10 & 3 \\ 0 & 11 \\ \end{array}\right]$$
- $S \Lambda S^{-1}$
  $$S = \left[ \begin{array}{rr} 3 & 1 \\ 1 & -1 \\ \end{array}\right];\quad \Lambda = \left[ \begin{array}{rr} -11 & 0 \\ 0 & 10 \\ \end{array}\right];\quad S^{-1} = \left[ \begin{array}{rr} \frac{2}{7} & \frac{1}{7} \\ \frac{1}{7} & -\frac{3}{7} \\ \end{array}\right]$$
- Przykład 4 — $A_{2x2}$
  $$A = \left[ \begin{array}{rr} -4 & 4 \\ 3 & 7 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -4 & 4 \\ 0 & 10 \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -4 & 0 \\ 0 & 10 \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & -1 \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & \frac{3}{5} \\ \frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 5 & 0 \\ 0 & 8 \\ \end{array}\right]$$
- $S \Lambda S^{-1}$
  $$S = \left[ \begin{array}{rr} -4 & 1 \\ 1 & 3 \\ \end{array}\right];\quad \Lambda = \left[ \begin{array}{rr} -5 & 0 \\ 0 & 8 \\ \end{array}\right];\quad S^{-1} = \left[ \begin{array}{rr} -\frac{3}{13} & \frac{1}{13} \\ \frac{1}{13} & \frac{4}{13} \\ \end{array}\right]$$
- Przykład 5 — $A_{2x2}$
  $$A = \left[ \begin{array}{rr} -8 & -7 \\ 6 & 9 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -8 & -7 \\ 0 & \frac{15}{4} \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -8 & 0 \\ 0 & \frac{15}{4} \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & \frac{7}{8} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & \frac{3}{5} \\ \frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 10 & 11 \\ 0 & 3 \\ \end{array}\right]$$
- $S \Lambda S^{-1}$
  $$S = \left[ \begin{array}{rr} -\frac{7}{3} & 1 \\ 1 & -2 \\ \end{array}\right];\quad \Lambda = \left[ \begin{array}{rr} -5 & 0 \\ 0 & 6 \\ \end{array}\right];\quad S^{-1} = \left[ \begin{array}{rr} -\frac{6}{11} & -\frac{3}{11} \\ -\frac{3}{11} & -\frac{7}{11} \\ \end{array}\right]$$
- Przykład 6 — $A_{2x2}$
  $$A = \left[ \begin{array}{rr} -4 & -6 \\ -3 & 3 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -4 & -6 \\ 0 & \frac{15}{2} \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -4 & 0 \\ 0 & \frac{15}{2} \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & \frac{3}{2} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & -\frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 5 & 3 \\ 0 & 6 \\ \end{array}\right]$$
- Przykład 7 — $A_{2x2}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -8 & 6 \\ 6 & -2 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -8 & 6 \\ 0 & \frac{5}{2} \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -8 & 0 \\ 0 & \frac{5}{2} \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & -\frac{3}{4} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & \frac{3}{5} \\ \frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 10 & -6 \\ 0 & 2 \\ \end{array}\right]$$
- Przykład 8 — $A_{2x2}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -4 & -3 \\ -3 & 9 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -4 & -3 \\ 0 & \frac{45}{4} \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -4 & 0 \\ 0 & \frac{45}{4} \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & \frac{3}{4} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & -\frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 5 & -3 \\ 0 & 9 \\ \end{array}\right]$$
- Przykład 9 — $A_{2x2}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -8 & -6 \\ -6 & 8 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -8 & -6 \\ 0 & \frac{25}{2} \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -8 & 0 \\ 0 & \frac{25}{2} \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & \frac{3}{4} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & -\frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 10 & 0 \\ 0 & 10 \\ \end{array}\right]$$
- $S \Lambda S^{-1}$
  $$S = \left[ \begin{array}{rr} 3 & 1 \\ 1 & -3 \\ \end{array}\right];\quad \Lambda = \left[ \begin{array}{rr} -10 & 0 \\ 0 & 10 \\ \end{array}\right];\quad S^{-1} = \left[ \begin{array}{rr} \frac{3}{10} & \frac{1}{10} \\ \frac{1}{10} & -\frac{3}{10} \\ \end{array}\right]$$
- $Q \Lambda Q^{T}$
  $$Q = S\cdot \left[ \begin{array}{rr} \frac{1}{\sqrt{10}} & 0 \\ 0 & \frac{1}{\sqrt{10}} \\ \end{array}\right]$$
- Przykład 10 — $A_{2x2}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -4 & -3 \\ -3 & 4 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -4 & -3 \\ 0 & \frac{25}{4} \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ \frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -4 & 0 \\ 0 & \frac{25}{4} \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & \frac{3}{4} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & -\frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 5 & 0 \\ 0 & 5 \\ \end{array}\right]$$
- $S \Lambda S^{-1}$
  $$S = \left[ \begin{array}{rr} 3 & 1 \\ 1 & -3 \\ \end{array}\right];\quad \Lambda = \left[ \begin{array}{rr} -5 & 0 \\ 0 & 5 \\ \end{array}\right];\quad S^{-1} = \left[ \begin{array}{rr} \frac{3}{10} & \frac{1}{10} \\ \frac{1}{10} & -\frac{3}{10} \\ \end{array}\right]$$
- $Q \Lambda Q^{T}$
  $$Q = S\cdot \left[ \begin{array}{rr} \frac{1}{\sqrt{10}} & 0 \\ 0 & \frac{1}{\sqrt{10}} \\ \end{array}\right]$$
- Przykład 11 — $A_{2x2}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -8 & 6 \\ 6 & 8 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -8 & 6 \\ 0 & \frac{25}{2} \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -8 & 0 \\ 0 & \frac{25}{2} \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & -\frac{3}{4} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & \frac{3}{5} \\ \frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 10 & 0 \\ 0 & 10 \\ \end{array}\right]$$
- $S \Lambda S^{-1}$
  $$S = \left[ \begin{array}{rr} -3 & 1 \\ 1 & 3 \\ \end{array}\right];\quad \Lambda = \left[ \begin{array}{rr} -10 & 0 \\ 0 & 10 \\ \end{array}\right];\quad S^{-1} = \left[ \begin{array}{rr} -\frac{3}{10} & \frac{1}{10} \\ \frac{1}{10} & \frac{3}{10} \\ \end{array}\right]$$
- $Q \Lambda Q^{T}$
  $$Q = S\cdot \left[ \begin{array}{rr} \frac{1}{\sqrt{10}} & 0 \\ 0 & \frac{1}{\sqrt{10}} \\ \end{array}\right]$$
- Przykład 12 — $A_{2x2}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -8 & 6 \\ 6 & 8 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{3}{4} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -8 & 6 \\ 0 & \frac{25}{2} \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 \\ -\frac{3}{4} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -8 & 0 \\ 0 & \frac{25}{2} \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & -\frac{3}{4} \\ 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & \frac{3}{5} \\ \frac{3}{5} & \frac{4}{5} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 10 & 0 \\ 0 & 10 \\ \end{array}\right]$$

macierze 3x3

- Przykład 1 — $A_{3x3}$
  $$A = \left[ \begin{array}{rr} -6 & -2 & -7 \\ -6 & 5 & 9 \\ -3 & 0 & 2 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ 1 & 1 & 0 \\ \frac{1}{2} & \frac{1}{7} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -6 & -2 & -7 \\ 0 & 7 & 16 \\ 0 & 0 & \frac{45}{14} \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ 1 & 1 & 0 \\ \frac{1}{2} & \frac{1}{7} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -6 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & \frac{45}{14} \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & \frac{1}{3} & \frac{7}{6} \\ 0 & 1 & \frac{16}{7} \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{2}{3} & -\frac{2}{3} & -\frac{1}{3} \\ -\frac{2}{3} & \frac{11}{15} & -\frac{2}{15} \\ -\frac{1}{3} & -\frac{2}{15} & \frac{14}{15} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 9 & -2 & -2 \\ 0 & 5 & 11 \\ 0 & 0 & 3 \\ \end{array}\right]$$
- Przykład 2 — $A_{3x3}$
  $$A = \left[ \begin{array}{rr} -2 & -5 & -9 \\ 2 & 0 & -6 \\ -1 & -5 & -3 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ \frac{1}{2} & \frac{1}{2} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -2 & -5 & -9 \\ 0 & -5 & -15 \\ 0 & 0 & 9 \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ \frac{1}{2} & \frac{1}{2} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -2 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 9 \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & \frac{5}{2} & \frac{9}{2} \\ 0 & 1 & 3 \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{2}{3} & -\frac{1}{3} & -\frac{2}{3} \\ \frac{2}{3} & -\frac{2}{3} & -\frac{1}{3} \\ -\frac{1}{3} & -\frac{2}{3} & \frac{2}{3} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 3 & 5 & 3 \\ 0 & 5 & 9 \\ 0 & 0 & 6 \\ \end{array}\right]$$
- Przykład 3 — $A_{3x3}$
  $$A = \left[ \begin{array}{rr} -7 & 3 & -1 \\ 6 & 9 & -5 \\ 6 & 0 & 2 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -\frac{6}{7} & 1 & 0 \\ -\frac{6}{7} & \frac{2}{9} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -7 & 3 & -1 \\ 0 & \frac{81}{7} & -\frac{41}{7} \\ 0 & 0 & \frac{22}{9} \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -\frac{6}{7} & 1 & 0 \\ -\frac{6}{7} & \frac{2}{9} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -7 & 0 & 0 \\ 0 & \frac{81}{7} & 0 \\ 0 & 0 & \frac{22}{9} \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & -\frac{3}{7} & \frac{1}{7} \\ 0 & 0 & -\frac{41}{81} \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{7}{11} & \frac{6}{11} & \frac{6}{11} \\ \frac{6}{11} & \frac{9}{11} & -\frac{2}{11} \\ \frac{6}{11} & -\frac{2}{11} & \frac{9}{11} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 11 & 3 & 0 \\ 0 & 9 & -5 \\ 0 & 0 & 2 \\ \end{array}\right]$$
- Przykład 4 — $A_{3x3}$
  $$A = \left[ \begin{array}{rr} -8 & 5 & 5 \\ -6 & 5 & -5 \\ 0 & 0 & 5 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ \frac{3}{4} & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -8 & 5 & 5 \\ 0 & \frac{5}{4} & -\frac{35}{4} \\ 0 & 0 & 5 \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ \frac{3}{4} & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -8 & 0 & 0 \\ 0 & \frac{5}{4} & 0 \\ 0 & 0 & 5 \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & -\frac{5}{8} & -\frac{5}{8} \\ 0 & 1 & -7 \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{4}{5} & -\frac{3}{5} & 0 \\ -\frac{3}{5} & \frac{4}{5} & 0 \\ 0 & 0 & 1 \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 10 & -7 & -1 \\ 0 & 1 & -7 \\ 0 & 0 & 5 \\ \end{array}\right]$$
- Przykład 5 — $A_{3x3}$
  $$A = \left[ \begin{array}{rr} -2 & -4 & 5 \\ 1 & -3 & 5 \\ -2 & 1 & 5 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -\frac{1}{2} & 1 & 0 \\ 1 & -1 & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -2 & -4 & 5 \\ 0 & -5 & \frac{15}{2} \\ 0 & 0 & \frac{15}{2} \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -\frac{1}{2} & 1 & 0 \\ 1 & -1 & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -2 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & \frac{15}{2} \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & 2 & -\frac{5}{2} \\ 0 & 1 & -\frac{3}{2} \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{2}{3} & -\frac{2}{3} & -\frac{1}{3} \\ \frac{1}{3} & -\frac{2}{3} & \frac{2}{3} \\ -\frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 3 & 1 & -5 \\ 0 & 5 & -5 \\ 0 & 0 & 5 \\ \end{array}\right]$$
- Przykład 6 — $A_{3x3}$
  $$A = \left[ \begin{array}{rr} -6 & -8 & 2 \\ 6 & -1 & 2 \\ 3 & -5 & 6 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ -\frac{1}{2} & 1 & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -6 & -8 & 2 \\ 0 & -9 & 4 \\ 0 & 0 & 3 \\ \end{array}\right]$$
- $LDU'$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ -\frac{1}{2} & 1 & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -6 & 0 & 0 \\ 0 & -9 & 0 \\ 0 & 0 & 3 \\ \end{array}\right];\quad U' = \left[ \begin{array}{rr} 1 & \frac{4}{3} & -\frac{1}{3} \\ 0 & 1 & -\frac{4}{9} \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{2}{3} & -\frac{2}{3} & -\frac{1}{3} \\ \frac{2}{3} & -\frac{1}{3} & -\frac{2}{3} \\ \frac{1}{3} & -\frac{2}{3} & \frac{2}{3} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 9 & 3 & 2 \\ 0 & 9 & -6 \\ 0 & 0 & 2 \\ \end{array}\right]$$
- Przykład 7 — $A_{3x3}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -2 & 2 & -1 \\ 2 & -4 & 0 \\ -1 & 0 & 2 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ \frac{1}{2} & \frac{1}{2} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -2 & 2 & -1 \\ 0 & -2 & -1 \\ 0 & 0 & 3 \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ \frac{1}{2} & \frac{1}{2} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 3 \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & -1 & \frac{1}{2} \\ 0 & 1 & \frac{1}{2} \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{2}{3} & -\frac{1}{3} & -\frac{2}{3} \\ \frac{2}{3} & -\frac{2}{3} & -\frac{1}{3} \\ -\frac{1}{3} & -\frac{2}{3} & \frac{2}{3} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 3 & -4 & 0 \\ 0 & 2 & -1 \\ 0 & 0 & 2 \\ \end{array}\right]$$
- Przykład 8 — $A_{3x3}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -8 & -4 & 8 \\ -4 & -7 & 9 \\ 8 & 9 & 2 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ \frac{1}{2} & 1 & 0 \\ -1 & -1 & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -8 & -4 & 8 \\ 0 & -5 & 5 \\ 0 & 0 & 15 \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ \frac{1}{2} & 1 & 0 \\ -1 & -1 & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -8 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 15 \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & \frac{1}{2} & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{2}{3} & \frac{2}{3} & \frac{1}{3} \\ -\frac{1}{3} & -\frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 12 & 11 & -7 \\ 0 & 5 & 0 \\ 0 & 0 & 10 \\ \end{array}\right]$$
- Przykład 9 — $A_{3x3}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -4 & -4 & -2 \\ -4 & -8 & 0 \\ -2 & 0 & 7 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ 1 & 1 & 0 \\ \frac{1}{2} & -\frac{1}{2} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -4 & -4 & -2 \\ 0 & -4 & 2 \\ 0 & 0 & 9 \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ 1 & 1 & 0 \\ \frac{1}{2} & -\frac{1}{2} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -4 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 9 \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & 1 & \frac{1}{2} \\ 0 & 1 & -\frac{1}{2} \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{2}{3} & \frac{1}{3} & -\frac{2}{3} \\ -\frac{2}{3} & -\frac{2}{3} & \frac{1}{3} \\ -\frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 6 & 8 & -1 \\ 0 & 4 & 4 \\ 0 & 0 & 6 \\ \end{array}\right]$$
- Przykład 10 — $A_{3x3}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -4 & 4 & 2 \\ 4 & -8 & 0 \\ 2 & 0 & 1 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ -\frac{1}{2} & -\frac{1}{2} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -4 & 4 & 2 \\ 0 & -4 & 2 \\ 0 & 0 & 3 \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ -\frac{1}{2} & -\frac{1}{2} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -4 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 3 \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & -1 & -\frac{1}{2} \\ 0 & 1 & -\frac{1}{2} \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{2}{3} & -\frac{1}{3} & \frac{2}{3} \\ \frac{2}{3} & -\frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 6 & -8 & -1 \\ 0 & 4 & 0 \\ 0 & 0 & 2 \\ \end{array}\right]$$
- Przykład 11 — $A_{3x3}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -7 & -4 & -4 \\ -4 & 0 & -2 \\ -4 & -2 & 9 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ \frac{4}{7} & 1 & 0 \\ \frac{4}{7} & \frac{1}{8} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -7 & -4 & -4 \\ 0 & \frac{16}{7} & \frac{2}{7} \\ 0 & 0 & \frac{45}{4} \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ \frac{4}{7} & 1 & 0 \\ \frac{4}{7} & \frac{1}{8} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -7 & 0 & 0 \\ 0 & \frac{16}{7} & 0 \\ 0 & 0 & \frac{45}{4} \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & \frac{4}{7} & \frac{4}{7} \\ 0 & 1 & \frac{1}{8} \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{7}{9} & -\frac{4}{9} & -\frac{4}{9} \\ -\frac{4}{9} & \frac{8}{9} & -\frac{1}{9} \\ -\frac{4}{9} & -\frac{1}{9} & \frac{8}{9} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 9 & 4 & 0 \\ 0 & 2 & -1 \\ 0 & 0 & 10 \\ \end{array}\right]$$
- Przykład 12 — $A_{3x3}$ [m. symetryczna]
  $$A = \left[ \begin{array}{rr} -6 & 3 & -6 \\ 3 & 4 & 2 \\ -6 & 2 & 1 \\ \end{array}\right]$$
- $LU$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -\frac{1}{2} & 1 & 0 \\ 1 & -\frac{2}{11} & 1 \\ \end{array}\right];\quad U = \left[ \begin{array}{rr} -6 & 3 & -6 \\ 0 & \frac{11}{2} & -1 \\ 0 & 0 & \frac{75}{11} \\ \end{array}\right]$$
- $LDU'=LDL^T$
  $$L = \left[ \begin{array}{rr} 1 & 0 & 0 \\ -\frac{1}{2} & 1 & 0 \\ 1 & -\frac{2}{11} & 1 \\ \end{array}\right];\quad D = \left[ \begin{array}{rr} -6 & 0 & 0 \\ 0 & \frac{11}{2} & 0 \\ 0 & 0 & \frac{75}{11} \\ \end{array}\right];\quad U'=L^T = \left[ \begin{array}{rr} 1 & -\frac{1}{2} & 1 \\ 0 & 1 & -\frac{2}{11} \\ 0 & 0 & 1 \\ \end{array}\right]$$
- $QR$
  $$Q = \left[ \begin{array}{rr} -\frac{2}{3} & \frac{1}{3} & -\frac{2}{3} \\ \frac{1}{3} & \frac{14}{15} & \frac{2}{15} \\ -\frac{2}{3} & \frac{2}{15} & \frac{11}{15} \\ \end{array}\right];\quad R = \left[ \begin{array}{rr} 9 & -2 & 4 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \\ \end{array}\right]$$