$$$\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]$$$:n singulaariarvohajotelma

Määritä matriisin $$$\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]$$$ SVD.

Laske matriisin transpoosi: $$$\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]^{T} = \left[\begin{array}{cc}t & 0\\- t & t\end{array}\right]$$$ (vaiheet: katso matriisin transpoosilaskin).

Kerro matriisi transpoosillaan: $$$W = \left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]\cdot \left[\begin{array}{cc}t & 0\\- t & t\end{array}\right] = \left[\begin{array}{cc}2 t^{2} & - t^{2}\\- t^{2} & t^{2}\end{array}\right]$$$ (vaiheiden näkemiseksi katso matriisikertolaskin).

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

Ominaisarvo: $$$\frac{t^{2} \left(3 - \sqrt{5}\right)}{2}$$$, ominaisvektori: $$$\left[\begin{array}{c}\frac{-1 + \sqrt{5}}{2}\\1\end{array}\right]$$$.

Ominaisarvo: $$$\frac{t^{2} \left(\sqrt{5} + 3\right)}{2}$$$, ominaisvektori: $$$\left[\begin{array}{c}- \frac{1 + \sqrt{5}}{2}\\1\end{array}\right]$$$.

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

$$$\sigma_{1} = \frac{\sqrt{2} \sqrt{3 - \sqrt{5}} \left|{t}\right|}{2}$$$

$$$\sigma_{2} = \frac{\sqrt{2} \sqrt{\sqrt{5} + 3} \left|{t}\right|}{2}$$$

Matriisi $$$\Sigma$$$ on nollamatriisi, jonka diagonaalilla on $$$\sigma_{i}$$$: $$$\Sigma = \left[\begin{array}{cc}\frac{\sqrt{2} \sqrt{3 - \sqrt{5}} \left|{t}\right|}{2} & 0\\0 & \frac{\sqrt{2} \sqrt{\sqrt{5} + 3} \left|{t}\right|}{2}\end{array}\right].$$$

Matriisin $$$U$$$ sarakkeet ovat normalisoidut (yksikkö)vektorit: $$$U = \left[\begin{array}{cc}\frac{- \sqrt{2} + \sqrt{10}}{2 \sqrt{5 - \sqrt{5}}} & - \frac{\sqrt{2} + \sqrt{10}}{2 \sqrt{\sqrt{5} + 5}}\\\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}} & \frac{\sqrt{2}}{\sqrt{\sqrt{5} + 5}}\end{array}\right]$$$ (yksikkövektorin löytämisen vaiheet, katso yksikkövektorilaskin).

Nyt, $$$v_{i} = \frac{1}{\sigma_{i}}\cdot \left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]^{T}\cdot u_{i}$$$:

$$$v_{1} = \frac{1}{\sigma_{1}}\cdot \left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]^{T}\cdot u_{1} = \frac{1}{\frac{\sqrt{2} \sqrt{3 - \sqrt{5}} \left|{t}\right|}{2}}\cdot \left[\begin{array}{cc}t & 0\\- t & t\end{array}\right]\cdot \left[\begin{array}{c}\frac{- \sqrt{2} + \sqrt{10}}{2 \sqrt{5 - \sqrt{5}}}\\\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}\end{array}\right] = \left[\begin{array}{c}\frac{t \left(-1 + \sqrt{5}\right)}{2 \sqrt{5 - 2 \sqrt{5}} \left|{t}\right|}\\\frac{t \left(3 - \sqrt{5}\right)}{2 \sqrt{5 - 2 \sqrt{5}} \left|{t}\right|}\end{array}\right]$$$ (vaiheista katso matriisin skalaarikertolaskin ja matriisikertolaskin).

$$$v_{2} = \frac{1}{\sigma_{2}}\cdot \left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]^{T}\cdot u_{2} = \frac{1}{\frac{\sqrt{2} \sqrt{\sqrt{5} + 3} \left|{t}\right|}{2}}\cdot \left[\begin{array}{cc}t & 0\\- t & t\end{array}\right]\cdot \left[\begin{array}{c}- \frac{\sqrt{2} + \sqrt{10}}{2 \sqrt{\sqrt{5} + 5}}\\\frac{\sqrt{2}}{\sqrt{\sqrt{5} + 5}}\end{array}\right] = \left[\begin{array}{c}- \frac{t \left(1 + \sqrt{5}\right)}{2 \sqrt{2 \sqrt{5} + 5} \left|{t}\right|}\\\frac{t \left(\sqrt{5} + 3\right)}{2 \sqrt{2 \sqrt{5} + 5} \left|{t}\right|}\end{array}\right]$$$ (vaiheista katso matriisin skalaarikertolaskin ja matriisikertolaskin).

Siispä $$$V = \left[\begin{array}{cc}\frac{t \left(-1 + \sqrt{5}\right)}{2 \sqrt{5 - 2 \sqrt{5}} \left|{t}\right|} & - \frac{t \left(1 + \sqrt{5}\right)}{2 \sqrt{2 \sqrt{5} + 5} \left|{t}\right|}\\\frac{t \left(3 - \sqrt{5}\right)}{2 \sqrt{5 - 2 \sqrt{5}} \left|{t}\right|} & \frac{t \left(\sqrt{5} + 3\right)}{2 \sqrt{2 \sqrt{5} + 5} \left|{t}\right|}\end{array}\right].$$$

Matriisit $$$U$$$, $$$\Sigma$$$ ja $$$V$$$ ovat sellaiset, että alkuperäinen matriisi $$$\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right] = U \Sigma V^T$$$.

$$$U = \left[\begin{array}{cc}\frac{- \sqrt{2} + \sqrt{10}}{2 \sqrt{5 - \sqrt{5}}} & - \frac{\sqrt{2} + \sqrt{10}}{2 \sqrt{\sqrt{5} + 5}}\\\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}} & \frac{\sqrt{2}}{\sqrt{\sqrt{5} + 5}}\end{array}\right]\approx \left[\begin{array}{cc}0.525731112119134 & -0.85065080835204\\0.85065080835204 & 0.525731112119134\end{array}\right]$$$A

$$$\Sigma = \left[\begin{array}{cc}\frac{\sqrt{2} \sqrt{3 - \sqrt{5}} \left|{t}\right|}{2} & 0\\0 & \frac{\sqrt{2} \sqrt{\sqrt{5} + 3} \left|{t}\right|}{2}\end{array}\right]\approx \left[\begin{array}{cc}0.618033988749895 \left|{t}\right| & 0\\0 & 1.618033988749895 \left|{t}\right|\end{array}\right]$$$A

$$$V = \left[\begin{array}{cc}\frac{t \left(-1 + \sqrt{5}\right)}{2 \sqrt{5 - 2 \sqrt{5}} \left|{t}\right|} & - \frac{t \left(1 + \sqrt{5}\right)}{2 \sqrt{2 \sqrt{5} + 5} \left|{t}\right|}\\\frac{t \left(3 - \sqrt{5}\right)}{2 \sqrt{5 - 2 \sqrt{5}} \left|{t}\right|} & \frac{t \left(\sqrt{5} + 3\right)}{2 \sqrt{2 \sqrt{5} + 5} \left|{t}\right|}\end{array}\right]\approx \left[\begin{array}{cc}\frac{0.85065080835204 t}{\left|{t}\right|} & - \frac{0.525731112119134 t}{\left|{t}\right|}\\\frac{0.525731112119134 t}{\left|{t}\right|} & \frac{0.85065080835204 t}{\left|{t}\right|}\end{array}\right]$$$A