SVD of $$$\left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]$$$

Find the SVD of $$$\left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]$$$.

Find the transpose of the matrix: $$$\left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]^{T} = \left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]$$$ (for steps, see matrix transpose calculator).

Multiply the matrix with its transpose: $$$W = \left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]\cdot \left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right] = \left[\begin{array}{cc}\frac{2}{3} & 0\\0 & \frac{1}{3}\end{array}\right]$$$ (for steps, see matrix multiplication calculator).

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

Eigenvalue: $$$\frac{2}{3}$$$, eigenvector: $$$\left[\begin{array}{c}1\\0\end{array}\right]$$$.

Eigenvalue: $$$\frac{1}{3}$$$, eigenvector: $$$\left[\begin{array}{c}0\\1\end{array}\right]$$$.

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

$$$\sigma_{1} = \frac{\sqrt{6}}{3}$$$

$$$\sigma_{2} = \frac{\sqrt{3}}{3}$$$

The $$$\Sigma$$$ matrix is a zero matrix with $$$\sigma_{i}$$$ on its diagonal: $$$\Sigma = \left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]$$$.

The columns of the matrix $$$U$$$ are the normalized (unit) vectors: $$$U = \left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right]$$$ (for steps in finding a unit vector, see unit vector calculator).

Now, $$$v_{i} = \frac{1}{\sigma_{i}}\cdot \left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]^{T}\cdot u_{i}$$$:

$$$v_{1} = \frac{1}{\sigma_{1}}\cdot \left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]^{T}\cdot u_{1} = \frac{1}{\frac{\sqrt{6}}{3}}\cdot \left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]\cdot \left[\begin{array}{c}1\\0\end{array}\right] = \left[\begin{array}{c}1\\0\end{array}\right]$$$ (for steps, see matrix scalar multiplication calculator and matrix multiplication calculator).

$$$v_{2} = \frac{1}{\sigma_{2}}\cdot \left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]^{T}\cdot u_{2} = \frac{1}{\frac{\sqrt{3}}{3}}\cdot \left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]\cdot \left[\begin{array}{c}0\\1\end{array}\right] = \left[\begin{array}{c}0\\1\end{array}\right]$$$ (for steps, see matrix scalar multiplication calculator and matrix multiplication calculator).

Therefore, $$$V = \left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right]$$$.

The matrices $$$U$$$, $$$\Sigma$$$, and $$$V$$$ are such that the initial matrix $$$\left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right] = U \Sigma V^T$$$.

$$$U = \left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right]$$$A

$$$\Sigma = \left[\begin{array}{cc}\frac{\sqrt{6}}{3} & 0\\0 & \frac{\sqrt{3}}{3}\end{array}\right]\approx \left[\begin{array}{cc}0.816496580927726 & 0\\0 & 0.577350269189626\end{array}\right]$$$A

$$$V = \left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right]$$$A