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

Find the SVD of $$$\left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]$$$.

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

Multiply the matrix with its transpose: $$$W = \left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]\cdot \left[\begin{array}{cc}2 \sqrt{2} & 2 \sqrt{2}\end{array}\right] = \left[\begin{array}{cc}8 & 8\\8 & 8\end{array}\right]$$$ (for steps, see matrix multiplication calculator).

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

Eigenvalue: $$$16$$$, eigenvector: $$$\left[\begin{array}{c}1\\1\end{array}\right]$$$.

Eigenvalue: $$$0$$$, eigenvector: $$$\left[\begin{array}{c}-1\\1\end{array}\right]$$$.

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

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

The $$$\Sigma$$$ matrix is a zero matrix with $$$\sigma_{i}$$$ on its diagonal: $$$\Sigma = \left[\begin{array}{c}4\\0\end{array}\right]$$$.

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

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

$$$v_{1} = \frac{1}{\sigma_{1}}\cdot \left[\begin{array}{c}2 \sqrt{2}\\2 \sqrt{2}\end{array}\right]^{T}\cdot u_{1} = \frac{1}{4}\cdot \left[\begin{array}{cc}2 \sqrt{2} & 2 \sqrt{2}\end{array}\right]\cdot \left[\begin{array}{c}\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{array}\right] = \left[\begin{array}{c}1\end{array}\right]$$$ (for steps, see matrix scalar multiplication calculator and matrix multiplication calculator).

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

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

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

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

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