SVD dari $$$\left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]$$$

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

Temukan transpos matriks: $$$\left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]^{T} = \left[\begin{array}{cc}2 \sqrt{2} & 2 \sqrt{2}\end{array}\right]$$$ (untuk langkah-langkahnya, lihat kalkulator transpos matriks).

Kalikan matriks dengan transposnya: $$$W = \left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]\cdot \left[\begin{array}{cc}2 \sqrt{2} & 2 \sqrt{2}\end{array}\right] = \left[\begin{array}{cc}8 & 8\\8 & 8\end{array}\right]$$$ (untuk langkah-langkahnya, lihat kalkulator perkalian matriks).

Sekarang, temukan nilai eigen dan vektor eigen dari $$$W$$$ (untuk langkah-langkahnya, lihat kalkulator nilai dan vektor eigen).

Nilai eigen: $$$16$$$, vektor eigen: $$$\left[\begin{array}{c}1\\1\end{array}\right]$$$.

Nilai eigen: $$$0$$$, vektor eigen: $$$\left[\begin{array}{c}-1\\1\end{array}\right]$$$.

Temukan akar kuadrat dari nilai eigen tak nol ($$$\sigma_{i}$$$):

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

Matriks $$$\Sigma$$$ adalah matriks nol dengan $$$\sigma_{i}$$$ pada diagonalnya: $$$\Sigma = \left[\begin{array}{c}4\\0\end{array}\right]$$$.

Kolom-kolom matriks $$$U$$$ adalah vektor-vektor yang dinormalisasi (vektor satuan): $$$U = \left[\begin{array}{cc}\frac{\sqrt{2}}{2} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{array}\right]$$$ (untuk langkah-langkah mencari vektor satuan, lihat kalkulator vektor satuan).

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

$$$v_{1} = \frac{1}{\sigma_{1}}\cdot \left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]^{T}\cdot u_{1} = \frac{1}{4}\cdot \left[\begin{array}{cc}2 \sqrt{2} & 2 \sqrt{2}\end{array}\right]\cdot \left[\begin{array}{c}\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{array}\right] = \left[\begin{array}{c}1\end{array}\right]$$$ (untuk langkah-langkah, lihat kalkulator perkalian skalar matriks dan kalkulator perkalian matriks).

Oleh karena itu, $$$V = \left[\begin{array}{c}1\end{array}\right]$$$.

Matriks $$$U$$$, $$$\Sigma$$$, dan $$$V$$$ sedemikian rupa sehingga matriks awal $$$\left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right] = U \Sigma V^T$$$.

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

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

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