Kalkulator Dekomposisi Nilai Singular

Temukan SVD dari $$$\left[\begin{array}{ccc}0 & 1 & 1\\\sqrt{2} & 2 & 0\\0 & 1 & 1\end{array}\right]$$$.

Temukan transpos matriks: $$$\left[\begin{array}{ccc}0 & 1 & 1\\\sqrt{2} & 2 & 0\\0 & 1 & 1\end{array}\right]^{T} = \left[\begin{array}{ccc}0 & \sqrt{2} & 0\\1 & 2 & 1\\1 & 0 & 1\end{array}\right]$$$ (untuk langkah-langkahnya, lihat kalkulator transpos matriks).

Kalikan matriks dengan transposnya: $$$W = \left[\begin{array}{ccc}0 & 1 & 1\\\sqrt{2} & 2 & 0\\0 & 1 & 1\end{array}\right]\cdot \left[\begin{array}{ccc}0 & \sqrt{2} & 0\\1 & 2 & 1\\1 & 0 & 1\end{array}\right] = \left[\begin{array}{ccc}2 & 2 & 2\\2 & 6 & 2\\2 & 2 & 2\end{array}\right]$$$ (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: $$$8$$$, vektor eigen: $$$\left[\begin{array}{c}1\\2\\1\end{array}\right]$$$.

Nilai eigen: $$$2$$$, vektor eigen: $$$\left[\begin{array}{c}1\\-1\\1\end{array}\right]$$$.

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

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

$$$\sigma_{1} = 2 \sqrt{2}$$$

$$$\sigma_{2} = \sqrt{2}$$$

Matriks $$$\Sigma$$$ adalah matriks nol dengan $$$\sigma_{i}$$$ pada diagonalnya: $$$\Sigma = \left[\begin{array}{ccc}2 \sqrt{2} & 0 & 0\\0 & \sqrt{2} & 0\\0 & 0 & 0\end{array}\right]$$$.

Kolom-kolom matriks $$$U$$$ adalah vektor-vektor yang dinormalisasi (vektor satuan): $$$U = \left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & \frac{\sqrt{3}}{3} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{6}}{3} & - \frac{\sqrt{3}}{3} & 0\\\frac{\sqrt{6}}{6} & \frac{\sqrt{3}}{3} & \frac{\sqrt{2}}{2}\end{array}\right]$$$ (untuk langkah-langkah mencari vektor satuan, lihat kalkulator vektor satuan).

Sekarang, $$$v_{i} = \frac{1}{\sigma_{i}}\cdot \left[\begin{array}{ccc}0 & 1 & 1\\\sqrt{2} & 2 & 0\\0 & 1 & 1\end{array}\right]^{T}\cdot u_{i}$$$:

$$$v_{1} = \frac{1}{\sigma_{1}}\cdot \left[\begin{array}{ccc}0 & 1 & 1\\\sqrt{2} & 2 & 0\\0 & 1 & 1\end{array}\right]^{T}\cdot u_{1} = \frac{1}{2 \sqrt{2}}\cdot \left[\begin{array}{ccc}0 & \sqrt{2} & 0\\1 & 2 & 1\\1 & 0 & 1\end{array}\right]\cdot \left[\begin{array}{c}\frac{\sqrt{6}}{6}\\\frac{\sqrt{6}}{3}\\\frac{\sqrt{6}}{6}\end{array}\right] = \left[\begin{array}{c}\frac{\sqrt{6}}{6}\\\frac{\sqrt{3}}{2}\\\frac{\sqrt{3}}{6}\end{array}\right]$$$ (untuk langkah-langkah, lihat kalkulator perkalian skalar matriks dan kalkulator perkalian matriks).

$$$v_{2} = \frac{1}{\sigma_{2}}\cdot \left[\begin{array}{ccc}0 & 1 & 1\\\sqrt{2} & 2 & 0\\0 & 1 & 1\end{array}\right]^{T}\cdot u_{2} = \frac{1}{\sqrt{2}}\cdot \left[\begin{array}{ccc}0 & \sqrt{2} & 0\\1 & 2 & 1\\1 & 0 & 1\end{array}\right]\cdot \left[\begin{array}{c}\frac{\sqrt{3}}{3}\\- \frac{\sqrt{3}}{3}\\\frac{\sqrt{3}}{3}\end{array}\right] = \left[\begin{array}{c}- \frac{\sqrt{3}}{3}\\0\\\frac{\sqrt{6}}{3}\end{array}\right]$$$ (untuk langkah-langkah, lihat kalkulator perkalian skalar matriks dan kalkulator perkalian matriks).

Karena kita sudah kehabisan $$$\sigma_{i}$$$ non-nol dan memerlukan satu vektor lagi, cari vektor yang ortogonal terhadap semua vektor yang telah ditemukan dengan mencari ruang nol dari matriks yang baris-barisnya adalah vektor-vektor tersebut: $$$\left[\begin{array}{c}\sqrt{2}\\-1\\1\end{array}\right]$$$ (untuk langkah-langkahnya, lihat kalkulator ruang nol).

Normalkan vektor: menjadi $$$\left[\begin{array}{c}\frac{\sqrt{2}}{2}\\- \frac{1}{2}\\\frac{1}{2}\end{array}\right]$$$, (untuk langkah-langkahnya, lihat kalkulator vektor satuan).

Oleh karena itu, $$$V = \left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & - \frac{\sqrt{3}}{3} & \frac{\sqrt{2}}{2}\\\frac{\sqrt{3}}{2} & 0 & - \frac{1}{2}\\\frac{\sqrt{3}}{6} & \frac{\sqrt{6}}{3} & \frac{1}{2}\end{array}\right].$$$

Matriks $$$U$$$, $$$\Sigma$$$, dan $$$V$$$ sedemikian rupa sehingga matriks awal $$$\left[\begin{array}{ccc}0 & 1 & 1\\\sqrt{2} & 2 & 0\\0 & 1 & 1\end{array}\right] = U \Sigma V^T$$$.

$$$U = \left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & \frac{\sqrt{3}}{3} & - \frac{\sqrt{2}}{2}\\\frac{\sqrt{6}}{3} & - \frac{\sqrt{3}}{3} & 0\\\frac{\sqrt{6}}{6} & \frac{\sqrt{3}}{3} & \frac{\sqrt{2}}{2}\end{array}\right]\approx \left[\begin{array}{ccc}0.408248290463863 & 0.577350269189626 & -0.707106781186548\\0.816496580927726 & -0.577350269189626 & 0\\0.408248290463863 & 0.577350269189626 & 0.707106781186548\end{array}\right]$$$A

$$$\Sigma = \left[\begin{array}{ccc}2 \sqrt{2} & 0 & 0\\0 & \sqrt{2} & 0\\0 & 0 & 0\end{array}\right]\approx \left[\begin{array}{ccc}2.82842712474619 & 0 & 0\\0 & 1.414213562373095 & 0\\0 & 0 & 0\end{array}\right]$$$A

$$$V = \left[\begin{array}{ccc}\frac{\sqrt{6}}{6} & - \frac{\sqrt{3}}{3} & \frac{\sqrt{2}}{2}\\\frac{\sqrt{3}}{2} & 0 & - \frac{1}{2}\\\frac{\sqrt{3}}{6} & \frac{\sqrt{6}}{3} & \frac{1}{2}\end{array}\right]\approx \left[\begin{array}{ccc}0.408248290463863 & -0.577350269189626 & 0.707106781186548\\0.866025403784439 & 0 & -0.5\\0.288675134594813 & 0.816496580927726 & 0.5\end{array}\right]$$$A