# interpretation of an operation using singular value decomposition

Say I have a frequency response of a $$2 \times 2$$ system at a fixed frequency: $$G(i \omega$$). It can be anything, for example $$\begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}$$. If I compute the SVD, I get the following: $$U = \begin{bmatrix} -0.53 & -0.85 \\ -0.85 & 0.53 \end{bmatrix}$$, $$V = \begin{bmatrix} -0.77 & 0.64 \\ 0.64 & -0.77 \end{bmatrix}$$ and $$\Sigma = \begin{bmatrix} 5.12 & 0 \\ 0 & 1.95 \end{bmatrix}$$. Usually $$\bar{u}$$ and $$\bar{v}$$ are defined as the first columns of $$U$$ and $$V$$, respectively. These two are related to "strongest" singular value, which is obvious from this example. However, I don't know how to interpret this computation: $$$$\bar{v} \cdot \bar{u}^T$$$$ $$$$\begin{bmatrix} -0.77 \\ -0.64 \end{bmatrix} \cdot \begin{bmatrix} -0.53 \\ -0.85 \end{bmatrix}^T = \begin{bmatrix} 0.40 & 0.65 \\ 0.34 & 0.55 \end{bmatrix}$$$$ If I do this, what am I exactly computing and what is the interpretation? The norm of the outcome is always equal to 1, I guess because U and V are unitary.

If $$A=U\Sigma V^*$$ is the SVD, $$\Sigma$$ is the diagonal matrix of singular values and $$u_k, v_k$$ are the columns of $$U,V$$ respectively then we have $$A = \sum_k \sigma_k u_k v^*_k$$.
Then if $$\sigma_1$$ is much bigger than the other $$\sigma_k$$s then $$\sigma_1 u_1 v^*_1$$ is the 'dominant' part of $$A$$.
Note that you probably need the Hermitian transpose not just the transpose if you are dealing with frequency response. (And what is this $$i$$ in $$G(i \omega)$$?)
• Regarding the parenthetical, $G(s)$ is usually used to refer to the transfer function (Laplace transoform) so that plugging in $s = i \omega$ (where $i^2 = -1$) gives us the Fourier transform, i.e. frequency response. Jun 21 '20 at 9:32
• @Omnomnomnom: Sorry, that was tongue-in-cheek; I am an electrical engineer, usually it is written $G(j \omega)$, and is typically complex, hence my eyebrow raise :-). Jun 21 '20 at 15:01