SVD of $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]$$$

Find the SVD of $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]$$$.

Find the transpose of the matrix: $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]^{T} = \left[\begin{array}{cc}1 & 3\\2 & 4\end{array}\right]$$$ (for steps, see matrix transpose calculator).

Multiply the matrix with its transpose: $$$W = \left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]\cdot \left[\begin{array}{cc}1 & 3\\2 & 4\end{array}\right] = \left[\begin{array}{cc}5 & 11\\11 & 25\end{array}\right]$$$ (for steps, see matrix multiplication calculator).

Now, find the eigenvalues and eigenvectors of $$$W$$$ (for steps, see eigenvalues and eigenvectors calculator).

Eigenvalue: $$$15 - \sqrt{221}$$$, eigenvector: $$$\left[\begin{array}{c}- \frac{10 + \sqrt{221}}{11}\\1\end{array}\right]$$$.

Eigenvalue: $$$\sqrt{221} + 15$$$, eigenvector: $$$\left[\begin{array}{c}\frac{-10 + \sqrt{221}}{11}\\1\end{array}\right]$$$.

Find the square roots of the nonzero eigenvalues ($$$\sigma_{i}$$$):

$$$\sigma_{1} = \sqrt{15 - \sqrt{221}}$$$

$$$\sigma_{2} = \sqrt{\sqrt{221} + 15}$$$

The $$$\Sigma$$$ matrix is a zero matrix with $$$\sigma_{i}$$$ on its diagonal: $$$\Sigma = \left[\begin{array}{cc}\sqrt{15 - \sqrt{221}} & 0\\0 & \sqrt{\sqrt{221} + 15}\end{array}\right].$$$

The columns of the matrix $$$U$$$ are the normalized (unit) vectors: $$$U = \left[\begin{array}{cc}- \frac{10 + \sqrt{221}}{\sqrt{20 \sqrt{221} + 442}} & \frac{- 10 \sqrt{2} + \sqrt{442}}{2 \sqrt{221 - 10 \sqrt{221}}}\\\frac{11}{\sqrt{20 \sqrt{221} + 442}} & \frac{11}{\sqrt{442 - 20 \sqrt{221}}}\end{array}\right]$$$ (for steps in finding a unit vector, see unit vector calculator).

Now, $$$v_{i} = \frac{1}{\sigma_{i}}\cdot \left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]^{T}\cdot u_{i}$$$:

$$$v_{1} = \frac{1}{\sigma_{1}}\cdot \left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]^{T}\cdot u_{1} = \frac{1}{\sqrt{15 - \sqrt{221}}}\cdot \left[\begin{array}{cc}1 & 3\\2 & 4\end{array}\right]\cdot \left[\begin{array}{c}- \frac{10 + \sqrt{221}}{\sqrt{20 \sqrt{221} + 442}}\\\frac{11}{\sqrt{20 \sqrt{221} + 442}}\end{array}\right] = \left[\begin{array}{c}\frac{- \sqrt{442} + 23 \sqrt{2}}{2 \sqrt{1105 - 71 \sqrt{221}}}\\\frac{- \sqrt{442} + 12 \sqrt{2}}{\sqrt{1105 - 71 \sqrt{221}}}\end{array}\right]$$$ (for steps, see matrix scalar multiplication calculator and matrix multiplication calculator).

$$$v_{2} = \frac{1}{\sigma_{2}}\cdot \left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]^{T}\cdot u_{2} = \frac{1}{\sqrt{\sqrt{221} + 15}}\cdot \left[\begin{array}{cc}1 & 3\\2 & 4\end{array}\right]\cdot \left[\begin{array}{c}\frac{- 10 \sqrt{2} + \sqrt{442}}{2 \sqrt{221 - 10 \sqrt{221}}}\\\frac{11}{\sqrt{442 - 20 \sqrt{221}}}\end{array}\right] = \left[\begin{array}{c}\frac{\left(221 \sqrt{2} + 23 \sqrt{442}\right) \sqrt{71 \sqrt{221} + 1105}}{442 \sqrt{321 - 20 \sqrt{221}} \left(\sqrt{221} + 15\right)}\\\frac{\sqrt{71 \sqrt{221} + 1105} \left(\frac{12 \sqrt{442}}{221} + \sqrt{2}\right)}{\sqrt{321 - 20 \sqrt{221}} \left(\sqrt{221} + 15\right)}\end{array}\right]$$$ (for steps, see matrix scalar multiplication calculator and matrix multiplication calculator).

Therefore, $$$V = \left[\begin{array}{cc}\frac{- \sqrt{442} + 23 \sqrt{2}}{2 \sqrt{1105 - 71 \sqrt{221}}} & \frac{\left(221 \sqrt{2} + 23 \sqrt{442}\right) \sqrt{71 \sqrt{221} + 1105}}{442 \sqrt{321 - 20 \sqrt{221}} \left(\sqrt{221} + 15\right)}\\\frac{- \sqrt{442} + 12 \sqrt{2}}{\sqrt{1105 - 71 \sqrt{221}}} & \frac{\sqrt{71 \sqrt{221} + 1105} \left(\frac{12 \sqrt{442}}{221} + \sqrt{2}\right)}{\sqrt{321 - 20 \sqrt{221}} \left(\sqrt{221} + 15\right)}\end{array}\right].$$$

The matrices $$$U$$$, $$$\Sigma$$$, and $$$V$$$ are such that the initial matrix $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right] = U \Sigma V^T$$$.

$$$U = \left[\begin{array}{cc}- \frac{10 + \sqrt{221}}{\sqrt{20 \sqrt{221} + 442}} & \frac{- 10 \sqrt{2} + \sqrt{442}}{2 \sqrt{221 - 10 \sqrt{221}}}\\\frac{11}{\sqrt{20 \sqrt{221} + 442}} & \frac{11}{\sqrt{442 - 20 \sqrt{221}}}\end{array}\right]\approx \left[\begin{array}{cc}-0.914514295677304 & 0.404553584833757\\0.404553584833757 & 0.914514295677304\end{array}\right]$$$A

$$$\Sigma = \left[\begin{array}{cc}\sqrt{15 - \sqrt{221}} & 0\\0 & \sqrt{\sqrt{221} + 15}\end{array}\right]\approx \left[\begin{array}{cc}0.365966190626258 & 0\\0 & 5.464985704219043\end{array}\right]$$$A

$$$V = \left[\begin{array}{cc}\frac{- \sqrt{442} + 23 \sqrt{2}}{2 \sqrt{1105 - 71 \sqrt{221}}} & \frac{\left(221 \sqrt{2} + 23 \sqrt{442}\right) \sqrt{71 \sqrt{221} + 1105}}{442 \sqrt{321 - 20 \sqrt{221}} \left(\sqrt{221} + 15\right)}\\\frac{- \sqrt{442} + 12 \sqrt{2}}{\sqrt{1105 - 71 \sqrt{221}}} & \frac{\sqrt{71 \sqrt{221} + 1105} \left(\frac{12 \sqrt{442}}{221} + \sqrt{2}\right)}{\sqrt{321 - 20 \sqrt{221}} \left(\sqrt{221} + 15\right)}\end{array}\right]\approx \left[\begin{array}{cc}0.817415560470363 & 0.576048436766321\\-0.576048436766321 & 0.817415560470363\end{array}\right]$$$A