Calculadora de fatoração QR
Encontre a decomposição QR de uma matriz passo a passo
A calculadora encontrará a fatoração QR da matriz dada $$$A$$$ , ou seja, uma matriz ortogonal (ou semi-ortogonal) $$$Q$$$ e uma matriz triangular superior $$$R$$$ que $$$A=QR$$$ , com as etapas mostradas.
Calculadora relacionada: Calculadora de Decomposição LU
Sua entrada
Encontre a fatoração QR de $$$\left[\begin{array}{ccc}1 & 3 & 5\\1 & 3 & 1\\2 & -1 & 7\end{array}\right]$$$.
Solução
Ortonormalize o conjunto de vetores formado pelas colunas da matriz dada: $$$\left\{\left[\begin{array}{c}\frac{\sqrt{6}}{6}\\\frac{\sqrt{6}}{6}\\\frac{\sqrt{6}}{3}\end{array}\right], \left[\begin{array}{c}\frac{\sqrt{3}}{3}\\\frac{\sqrt{3}}{3}\\- \frac{\sqrt{3}}{3}\end{array}\right], \left[\begin{array}{c}\frac{\sqrt{2}}{2}\\- \frac{\sqrt{2}}{2}\\0\end{array}\right]\right\}$$$ (para os passos, veja Calculadora de Gram-Schmidt).
As colunas da matriz $$$Q$$$ são os vetores ortonormalizados: $$$Q = \left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & \frac{\sqrt{3}}{3} & \frac{\sqrt{2}}{2}\\\frac{\sqrt{6}}{6} & \frac{\sqrt{3}}{3} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{6}}{3} & - \frac{\sqrt{3}}{3} & 0\end{array}\right].$$$
Encontre a transposição da matriz: $$$Q^{T} = \left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{3}\\\frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & - \frac{\sqrt{3}}{3}\\\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & 0\end{array}\right]$$$ (para conhecer as etapas, consulte calculadora de transposição de matrizes).
Por fim, $$$R = \left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{6} & \frac{\sqrt{6}}{3}\\\frac{\sqrt{3}}{3} & \frac{\sqrt{3}}{3} & - \frac{\sqrt{3}}{3}\\\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2} & 0\end{array}\right]\left[\begin{array}{ccc}1 & 3 & 5\\1 & 3 & 1\\2 & -1 & 7\end{array}\right] = \left[\begin{array}{ccc}\sqrt{6} & \frac{2 \sqrt{6}}{3} & \frac{10 \sqrt{6}}{3}\\0 & \frac{7 \sqrt{3}}{3} & - \frac{\sqrt{3}}{3}\\0 & 0 & 2 \sqrt{2}\end{array}\right]$$$ (para conhecer as etapas, consulte calculadora de multiplicação de matrizes).
Responder
$$$Q = \left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & \frac{\sqrt{3}}{3} & \frac{\sqrt{2}}{2}\\\frac{\sqrt{6}}{6} & \frac{\sqrt{3}}{3} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{6}}{3} & - \frac{\sqrt{3}}{3} & 0\end{array}\right]\approx \left[\begin{array}{ccc}0.408248290463863 & 0.577350269189626 & 0.707106781186548\\0.408248290463863 & 0.577350269189626 & -0.707106781186548\\0.816496580927726 & -0.577350269189626 & 0\end{array}\right]$$$A
$$$R = \left[\begin{array}{ccc}\sqrt{6} & \frac{2 \sqrt{6}}{3} & \frac{10 \sqrt{6}}{3}\\0 & \frac{7 \sqrt{3}}{3} & - \frac{\sqrt{3}}{3}\\0 & 0 & 2 \sqrt{2}\end{array}\right]\approx \left[\begin{array}{ccc}2.449489742783178 & 1.632993161855452 & 8.16496580927726\\0 & 4.04145188432738 & -0.577350269189626\\0 & 0 & 2.82842712474619\end{array}\right]$$$A