# 21.4: Properties of the Structured Singular Value

It is important to note that $$\mu$$ is a function that depends on the perturbation class $$\Delta_{0}$$ (sometimes, this function is denoted by $$\mu_{Delta_{0}}$$ to indicate this dependence). The following are useful properties of such a function.

1. $$\mu(M) \geq 0$$.

2. If $$\Delta_{0}=\{\lambda I \mid \lambda \in \mathbb{C}\}$$, then $$\mu(M)=\rho(M)$$, the spectral radius of $$M$$ (which is equal to the magnitude of the eigenvalue of $$M$$ with maximum magnitude).

3. If $$\Delta=\{\Delta_{0} \mid \Delta$$ is an arbitrary complex matrix} then $$\mu=\sigma_{\max }(M)$$, from which $$\sup _{\omega} \mu=\|M\|_{\infty}$$.

Property 2 shows that the spectral radius function is a particular $$\mu$$ function with respect to a perturbation class consisting of matrices of the form of scaled identity. Property 3 shows that the maximum singular value function is a particular $$\mu$$ function with respect to a perturbation class consisting of arbitrary norm bounded perturbations (no structural constraints).

4. If $$\Delta=\left\{\operatorname{diag}\left(\Delta_{1}, \ldots, \Delta_{n}\right) \mid \Delta_{i}\right.$$ complex}, then $$\mu(M)=\mu\left(D^{-1} M D\right)$$ for any $$D = \operatorname{diag}\left(d_{1}, \ldots, d_{n}\right),\left|d_{i}\right| \geq 0$$. The set of such scales is denoted $$\mathcal{D}$$.

This can be seen by noting that $$\operatorname{det}(I-A B)=\operatorname{det}(I-B A)$$, so that $$\operatorname{det}\left(I-D^{-1} M D \Delta\right)=\operatorname{det}(I-\left.M D \Delta D^{-1}\right)=\operatorname{det}(I-M \Delta)$$. The last equality arises since the diagonal matrices $$\Delta)$$ and $$D$$ commute.

5. If $$\Delta_{0}=\operatorname{diag}\left(\Delta_{1}, \ldots, \Delta_{n}\right), \Delta_{i}$$ complex, then $$\rho(M)<\mu(M)<\sigma_{\max }(M)$$.

This property follows from the following observation: If $$\Delta_{01} \subset \Delta_{02}, \text { then } \mu_{1} \leq \mu_{2}$$. It is clear that the class of perturbations consisting of scaled identity matrices is a subset of $$\Delta_{0}$$ which is a subset of the class of all unstructured perturbations.

6. From 4 and 5 we have that $$\mu(M)=\mu\left(D^{-1} M D\right) \leq \inf _{D \in \mathcal{D}} \sigma_{\max }\left(D^{-1} M D\right)$$