# Doubt on Nyquist Criterion for MIMO systems

I am studying MIMO control systems, and I have found that in the Nyquist stability criterion for MIMO system, one of the condition for stability is that:

$$det(I+L(j\omega))\neq 0$$

and also that

$$det(I+L(j\omega))$$ has a number of encirclements of the origin equal to the number of poles in the RHP.

but what is $$det(I+L(j\omega))$$?

In SISO systems I know that we have to look at the encirclements of $$L(j\omega)$$ around $$-1$$.

But why in MIMO we have to look at $$det(I+L(j\omega))$$?

I put here the reference in which I found this: link

Nyquist's stability theorem is actually given in terms of $$1+L(j\omega)$$, which is the denominator polynomial of the close-loop transfer function. For SISO system, close-loop stability is examined by looking at the encirclement of $$1+L(j\omega)$$ around $$0$$, which is simplified as looking at the encirclement of $$L(j\omega)$$ around $$-1$$. Now for MIMO systems, we cannot just use $$\det(L(j\omega))$$ because that will cause pole-zero cancellation, for example with the plant $$\mathrm{diag}\left(\frac{s-1}{s+3},\frac{1}{s-1}\right)$$.