Given some arbitrary real parameter $b \in \mathbb R$, suppose we have the matrix \begin{align*} M = \begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix} \end{align*} Following up a generic formula found here, I decompose \begin{align*} \begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix} = \frac 1 2 \begin{pmatrix} 2 & -b \\ b & 2 \end{pmatrix} + \frac 1 2 \begin{pmatrix} 0 & b \\ b & 0 \end{pmatrix} \end{align*} The singular values $w_1$ and $w_2$ are implicitly given by \begin{align*} \frac{ w_1 + w_2 }{2} = \frac 1 2 \sqrt{ 4 + b^2 }, \quad \quad \frac{ w_1 - w_2 }{2} = \frac 1 2 |b|. \end{align*} Then adding and subtracting these equations leads to \begin{align*} w_1 = \frac 1 2 \sqrt{ 4 + b^2 } + \frac 1 2 |b| \\ w_2 = \frac 1 2 \sqrt{ 4 + b^2 } - \frac 1 2 |b| \end{align*} I would like to know
- Whether that reasoning is correct. For instance, the referenced formula might break down in the case of a triangular matrix (e.g., in the link above, what happens if $E = 0$ ? ).
- Ideally: what is the geometric interpretation of the SVD of $A$?
