$$$\left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]$$$:n singulaariarvohajotelma

Määritä matriisin $$$\left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]$$$ SVD.

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

Kerro matriisi transpoosillaan: $$$W = \left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]\cdot \left[\begin{array}{cc}2 \sqrt{2} & 2 \sqrt{2}\end{array}\right] = \left[\begin{array}{cc}8 & 8\\8 & 8\end{array}\right]$$$ (vaiheiden näkemiseksi katso matriisikertolaskin).

Seuraavaksi etsi $$$W$$$:n ominaisarvot ja ominaisvektorit (vaiheittaiset ohjeet: katso ominaisarvojen ja -vektorien laskin).

Ominaisarvo: $$$16$$$, ominaisvektori: $$$\left[\begin{array}{c}1\\1\end{array}\right]$$$.

Ominaisarvo: $$$0$$$, ominaisvektori: $$$\left[\begin{array}{c}-1\\1\end{array}\right]$$$.

Etsi nollasta poikkeavien ominaisarvojen ($$$\sigma_{i}$$$) neliöjuuret:

$$$\sigma_{1} = 4$$$

Matriisi $$$\Sigma$$$ on nollamatriisi, jonka diagonaalilla on $$$\sigma_{i}$$$: $$$\Sigma = \left[\begin{array}{c}4\\0\end{array}\right]$$$.

Matriisin $$$U$$$ sarakkeet ovat normalisoidut (yksikkö)vektorit: $$$U = \left[\begin{array}{cc}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{array}\right]$$$ (yksikkövektorin löytämisen vaiheet, katso yksikkövektorilaskin).

Nyt, $$$v_{i} = \frac{1}{\sigma_{i}}\cdot \left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]^{T}\cdot u_{i}$$$:

$$$v_{1} = \frac{1}{\sigma_{1}}\cdot \left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]^{T}\cdot u_{1} = \frac{1}{4}\cdot \left[\begin{array}{cc}2 \sqrt{2} & 2 \sqrt{2}\end{array}\right]\cdot \left[\begin{array}{c}\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{array}\right] = \left[\begin{array}{c}1\end{array}\right]$$$ (vaiheista katso matriisin skalaarikertolaskin ja matriisikertolaskin).

Siispä $$$V = \left[\begin{array}{c}1\end{array}\right]$$$.

Matriisit $$$U$$$, $$$\Sigma$$$ ja $$$V$$$ ovat sellaiset, että alkuperäinen matriisi $$$\left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right] = U \Sigma V^T$$$.

$$$U = \left[\begin{array}{cc}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{array}\right]\approx \left[\begin{array}{cc}0.707106781186548 & -0.707106781186548\\0.707106781186548 & 0.707106781186548\end{array}\right]$$$A

$$$\Sigma = \left[\begin{array}{c}4\\0\end{array}\right]$$$A

$$$V = \left[\begin{array}{c}1\end{array}\right]$$$A