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 model form.