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Kalkulator Aljabar Linear
Selesaikan soal aljabar linear langkah demi langkah
Masukan Anda
Temukan SVD dari $$$\left[\begin{array}{cc}1 & 1\\0 & 1\end{array}\right]$$$.
Solusi
Temukan transpos matriks: $$$\left[\begin{array}{cc}1 & 1\\0 & 1\end{array}\right]^{T} = \left[\begin{array}{cc}1 & 0\\1 & 1\end{array}\right]$$$ (untuk langkah-langkahnya, lihat kalkulator transpos matriks).
Kalikan matriks dengan transposnya: $$$W = \left[\begin{array}{cc}1 & 1\\0 & 1\end{array}\right]\cdot \left[\begin{array}{cc}1 & 0\\1 & 1\end{array}\right] = \left[\begin{array}{cc}2 & 1\\1 & 1\end{array}\right]$$$ (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: $$$- \frac{-3 + \sqrt{5}}{2}$$$, vektor eigen: $$$\left[\begin{array}{c}- \frac{-1 + \sqrt{5}}{2}\\1\end{array}\right]$$$.
Nilai eigen: $$$\frac{\sqrt{5} + 3}{2}$$$, vektor eigen: $$$\left[\begin{array}{c}\frac{1 + \sqrt{5}}{2}\\1\end{array}\right]$$$.
Temukan akar kuadrat dari nilai eigen tak nol ($$$\sigma_{i}$$$):
$$$\sigma_{1} = \frac{\sqrt{2} \sqrt{3 - \sqrt{5}}}{2}$$$
$$$\sigma_{2} = \frac{\sqrt{2} \sqrt{\sqrt{5} + 3}}{2}$$$
Matriks $$$\Sigma$$$ adalah matriks nol dengan $$$\sigma_{i}$$$ pada diagonalnya: $$$\Sigma = \left[\begin{array}{cc}\frac{\sqrt{2} \sqrt{3 - \sqrt{5}}}{2} & 0\\0 & \frac{\sqrt{2} \sqrt{\sqrt{5} + 3}}{2}\end{array}\right].$$$
Kolom-kolom matriks $$$U$$$ adalah vektor-vektor yang dinormalisasi (vektor satuan): $$$U = \left[\begin{array}{cc}\frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}} & \frac{\sqrt{2} + \sqrt{10}}{2 \sqrt{\sqrt{5} + 5}}\\\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}} & \frac{\sqrt{2}}{\sqrt{\sqrt{5} + 5}}\end{array}\right]$$$ (untuk langkah-langkah mencari vektor satuan, lihat kalkulator vektor satuan).
Sekarang, $$$v_{i} = \frac{1}{\sigma_{i}}\cdot \left[\begin{array}{cc}1 & 1\\0 & 1\end{array}\right]^{T}\cdot u_{i}$$$:
$$$v_{1} = \frac{1}{\sigma_{1}}\cdot \left[\begin{array}{cc}1 & 1\\0 & 1\end{array}\right]^{T}\cdot u_{1} = \frac{1}{\frac{\sqrt{2} \sqrt{3 - \sqrt{5}}}{2}}\cdot \left[\begin{array}{cc}1 & 0\\1 & 1\end{array}\right]\cdot \left[\begin{array}{c}\frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}}\\\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}\end{array}\right] = \left[\begin{array}{c}\frac{1 - \sqrt{5}}{2 \sqrt{5 - 2 \sqrt{5}}}\\\frac{3 - \sqrt{5}}{2 \sqrt{5 - 2 \sqrt{5}}}\end{array}\right]$$$ (untuk langkah-langkah, lihat kalkulator perkalian skalar matriks dan kalkulator perkalian matriks).
$$$v_{2} = \frac{1}{\sigma_{2}}\cdot \left[\begin{array}{cc}1 & 1\\0 & 1\end{array}\right]^{T}\cdot u_{2} = \frac{1}{\frac{\sqrt{2} \sqrt{\sqrt{5} + 3}}{2}}\cdot \left[\begin{array}{cc}1 & 0\\1 & 1\end{array}\right]\cdot \left[\begin{array}{c}\frac{\sqrt{2} + \sqrt{10}}{2 \sqrt{\sqrt{5} + 5}}\\\frac{\sqrt{2}}{\sqrt{\sqrt{5} + 5}}\end{array}\right] = \left[\begin{array}{c}\frac{1 + \sqrt{5}}{2 \sqrt{2 \sqrt{5} + 5}}\\\frac{\sqrt{5} + 3}{2 \sqrt{2 \sqrt{5} + 5}}\end{array}\right]$$$ (untuk langkah-langkah, lihat kalkulator perkalian skalar matriks dan kalkulator perkalian matriks).
Oleh karena itu, $$$V = \left[\begin{array}{cc}\frac{1 - \sqrt{5}}{2 \sqrt{5 - 2 \sqrt{5}}} & \frac{1 + \sqrt{5}}{2 \sqrt{2 \sqrt{5} + 5}}\\\frac{3 - \sqrt{5}}{2 \sqrt{5 - 2 \sqrt{5}}} & \frac{\sqrt{5} + 3}{2 \sqrt{2 \sqrt{5} + 5}}\end{array}\right].$$$
Matriks $$$U$$$, $$$\Sigma$$$, dan $$$V$$$ sedemikian rupa sehingga matriks awal $$$\left[\begin{array}{cc}1 & 1\\0 & 1\end{array}\right] = U \Sigma V^T$$$.
Jawaban
$$$U = \left[\begin{array}{cc}\frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}} & \frac{\sqrt{2} + \sqrt{10}}{2 \sqrt{\sqrt{5} + 5}}\\\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}} & \frac{\sqrt{2}}{\sqrt{\sqrt{5} + 5}}\end{array}\right]\approx \left[\begin{array}{cc}-0.525731112119134 & 0.85065080835204\\0.85065080835204 & 0.525731112119134\end{array}\right]$$$A
$$$\Sigma = \left[\begin{array}{cc}\frac{\sqrt{2} \sqrt{3 - \sqrt{5}}}{2} & 0\\0 & \frac{\sqrt{2} \sqrt{\sqrt{5} + 3}}{2}\end{array}\right]\approx \left[\begin{array}{cc}0.618033988749895 & 0\\0 & 1.618033988749895\end{array}\right]$$$A
$$$V = \left[\begin{array}{cc}\frac{1 - \sqrt{5}}{2 \sqrt{5 - 2 \sqrt{5}}} & \frac{1 + \sqrt{5}}{2 \sqrt{2 \sqrt{5} + 5}}\\\frac{3 - \sqrt{5}}{2 \sqrt{5 - 2 \sqrt{5}}} & \frac{\sqrt{5} + 3}{2 \sqrt{2 \sqrt{5} + 5}}\end{array}\right]\approx \left[\begin{array}{cc}-0.85065080835204 & 0.525731112119134\\0.525731112119134 & 0.85065080835204\end{array}\right]$$$A