Funktion $$$\left[\begin{array}{cc}2 & 1\\3 & 4\end{array}\right]$$$ pseudoinverssi

Laske Moore-Penrose-pseudoinverssi matriisille $$$\left[\begin{array}{cc}2 & 1\\3 & 4\end{array}\right]$$$.

Matriisin $$$A$$$ pseudoinverssi on $$$A^{+} = A^{T} \left(A A^{T}\right)^{-1}$$$.

Laske matriisin transpoosi: $$$\left[\begin{array}{cc}2 & 1\\3 & 4\end{array}\right]^{T} = \left[\begin{array}{cc}2 & 3\\1 & 4\end{array}\right]$$$ (vaiheet: katso matriisin transpoosilaskin).

Kerro alkuperäinen matriisi sen transpoosilla:

$$$\left[\begin{array}{cc}2 & 1\\3 & 4\end{array}\right]\cdot \left[\begin{array}{cc}2 & 3\\1 & 4\end{array}\right] = \left[\begin{array}{cc}5 & 10\\10 & 25\end{array}\right]$$$ (ratkaisuvaiheet: katso matriisikertolaskin).

Etsi käänteismatriisi: $$$\left[\begin{array}{cc}5 & 10\\10 & 25\end{array}\right]^{-1} = \left[\begin{array}{cc}1 & - \frac{2}{5}\\- \frac{2}{5} & \frac{1}{5}\end{array}\right]$$$ (vaiheet, katso käänteismatriisilaskin).

Lopuksi kerro matriisit:

$$$\left[\begin{array}{cc}2 & 3\\1 & 4\end{array}\right]\cdot \left[\begin{array}{cc}1 & - \frac{2}{5}\\- \frac{2}{5} & \frac{1}{5}\end{array}\right] = \left[\begin{array}{cc}\frac{4}{5} & - \frac{1}{5}\\- \frac{3}{5} & \frac{2}{5}\end{array}\right]$$$ (ratkaisuvaiheet: katso matriisikertolaskin).

$$$\left[\begin{array}{cc}2 & 1\\3 & 4\end{array}\right]^{+} = \left[\begin{array}{cc}\frac{4}{5} & - \frac{1}{5}\\- \frac{3}{5} & \frac{2}{5}\end{array}\right] = \left[\begin{array}{cc}0.8 & -0.2\\-0.6 & 0.4\end{array}\right]$$$A