interpretation of an operation using singular value decomposition

Question

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: \begin{equation} \bar{v} \cdot \bar{u}^T\end{equation} \begin{equation} \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}\end{equation} 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.

Answer 1

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 $G(i \omega)$?)