Modal Form

Suppose that an (autonomous) LDS is diagonalizable by a matrix $$T$$. Then, we can define new coordinates for states by $$x = T\tilde{x}$$, so :

$$T\dot{\tilde{x}}=AT\tilde{x}$$

Since $$T$$ is invertible (by assumption of diagonalizability), we have that:

$$\dot{\tilde{x}} = T^{-1} A T \tilde{x} = \Lambda \tilde{x}$$

where $$\Lambda$$ is the diagonal matrix of eigenvalues, $$\Lambda = \text{diag}(\lambda_1, \lambda_2,\dots, \lambda_n)$$.

Note that in this new coordinate system, the system is decoupled; that is, it consists of $$n$$ independent modes so that:

$$\tilde{x}_i(t) = e^{\lambda_i t} \tilde{x}_i(0)$$. This is why it is called modal form.

What's the point of a Modal Form?
Diagonalization is a powerful conceptual tool. That is, it's not necessarily useful for doing calculations, but it's very useful for (i) showing things (see also below) and for analyzing systems. The system $$\dot{x} = A x$$ is easier to analyze in the new coordinate system $$\dot{\tilde{x}} = \Lambda \tilde{x}$$.

Simplifying expressions
The following are examples to how modal form simplifies experssions:

1) The resolvent $$(sI-A)^{-1}$$ is very easy to compute on this new system; it's not difficult to show that :

$$(sI-A)^{-1} = T \text{diag}\left( \frac{1}{s-\lambda_1}, \dots, \frac{1}{s-\lambda_n} \right)T^{-1}$$.

2) Computing the matrix powers (and hence the discrete-time solution) is also very easy; it can be easily shown that: $$A^k = T\text{diag}(\lambda_1^k, \dots, \lambda_{n}^k )T^{-1}$$

3) Computing the exponential (and hence the continuous-time solution) is also very simple: $$e^A = T\text{diag}(e^{\lambda_1}, \dots, e^{\lambda_n})T^{-1}$$

Interpreting and analyzing system
The simplifications in expressions allow us to interpret systems and understand the conditions under which they are stable. Examples are:

1) Since the continuous-time solution can be written as a linear combination of the modes, we can interpret the solution (in the original coordinate system) as :

$$x(t) = e^{tA}x(0) = \sum_{i=1}^n e^{\lambda_i t}(w_i^T x(0)) v_i$$

This leads to the following interpretations: (i) left eigenvectors decompose the initial state $$x(0)$$ into modal components $$w^T_i x(0)$$ --- as can be seen by evaluating the expression above for $$t=0$$. (ii) we can reconstruct state as a linear combination of (right) eigenvectors.

2) System stability (continuous): The interpretations above allow us to find which initial conditions $$x(0)$$ lead to a stable system. It can be shown that an autonomous system is stable, only if $$x(0) \in \text{span}\{v_1,\dots,v_s\}$$, where $$v_1, \dots, v_s$$ are the eigenvectors of $$A$$ associated with eigenvalues that have a negative real part.

3) System stability (discrete): The modal form also shows as that a discrete-time system is stable (for all $$x(0)$$) iff the magnitude of all the eigenvalues of $$A$$ is smaller than 1. (I.e, the spectral norm of $$A$$ is smaller than 1).