Writing a matrix as multiplication of orthogonal and diagonal matrices Given is the matrix A = \begin{pmatrix}
  5 & 3  \\
  0 & -4 \\
  \end{pmatrix}
I would like to write this matrix as the product A = $U S V^T$
whereas U and V both are orthogonal matrices and S is a diagonal matrix. What is the approach to find suitable matrices?

Given:
$$A = \begin{pmatrix}
  5 & 3  \\
  0 & -4 \\
  \end{pmatrix}$$
We want to find the Singular Value Decomposition (SVD) where $U$ and $V$ both are orthogonal matrices and $\Sigma$ is a diagonal matrix, such that:
$$A = U \cdot \Sigma \cdot V^T$$
We arrive at:
$$A = U \cdot \Sigma \cdot V^T = \begin{pmatrix} 5 & 3  \\  0 & -4 \\ \end{pmatrix} = \begin{pmatrix} \frac{2}{\sqrt{5}} & -\frac{1}{\sqrt{5}}  \\  -\frac{1}{\sqrt{5}} & -\frac{2}{\sqrt{5}} \\ \end{pmatrix} \cdot \begin{pmatrix} 2\sqrt{10} & 0 \\  0 & \sqrt{10} \\ \end{pmatrix} \cdot \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}  \\  \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \end{pmatrix} $$
