Calculadora Pseudoinversa

Encontre o pseudoinverso de Moore-Penrose de $$$\left[\begin{array}{ccc}3 & 2 & 2\\2 & 3 & -2\end{array}\right]$$$.

O pseudoinverso de uma matriz $$$A$$$ é $$$A^{+} = A^{T} \left(A A^{T}\right)^{-1}$$$.

Encontre a transposição da matriz: $$$\left[\begin{array}{ccc}3 & 2 & 2\\2 & 3 & -2\end{array}\right]^{T} = \left[\begin{array}{cc}3 & 2\\2 & 3\\2 & -2\end{array}\right]$$$ (para conhecer as etapas, consulte calculadora de transposição de matrizes).

Multiplique a matriz original por sua transposta:

$$$\left[\begin{array}{ccc}3 & 2 & 2\\2 & 3 & -2\end{array}\right]\cdot \left[\begin{array}{cc}3 & 2\\2 & 3\\2 & -2\end{array}\right] = \left[\begin{array}{cc}17 & 8\\8 & 17\end{array}\right]$$$ (para ver as etapas, consulte calculadora de multiplicação de matrizes).

Encontre a matriz inversa: $$$\left[\begin{array}{cc}17 & 8\\8 & 17\end{array}\right]^{-1} = \left[\begin{array}{cc}\frac{17}{225} & - \frac{8}{225}\\- \frac{8}{225} & \frac{17}{225}\end{array}\right]$$$ (para ver as etapas, consulte calculadora inversa de matrizes).

Finalmente, multiplique as matrizes:

$$$\left[\begin{array}{cc}3 & 2\\2 & 3\\2 & -2\end{array}\right]\cdot \left[\begin{array}{cc}\frac{17}{225} & - \frac{8}{225}\\- \frac{8}{225} & \frac{17}{225}\end{array}\right] = \left[\begin{array}{cc}\frac{7}{45} & \frac{2}{45}\\\frac{2}{45} & \frac{7}{45}\\\frac{2}{9} & - \frac{2}{9}\end{array}\right]$$$ (para ver as etapas, consulte calculadora de multiplicação de matrizes).

$$$\left[\begin{array}{ccc}3 & 2 & 2\\2 & 3 & -2\end{array}\right]^{+} = \left[\begin{array}{cc}\frac{7}{45} & \frac{2}{45}\\\frac{2}{45} & \frac{7}{45}\\\frac{2}{9} & - \frac{2}{9}\end{array}\right]\approx \left[\begin{array}{cc}0.155555555555556 & 0.044444444444444\\0.044444444444444 & 0.155555555555556\\0.222222222222222 & -0.222222222222222\end{array}\right]$$$A