# Why do we use linear transformation of states in linear control technique

Just to leave some space here I link to an introduction of linear transformation of state. How is it useful? Can nayone provide an example where it is indeed useful and has to be done? http://www.staff.ul.ie/burkem/Teaching/st-tr.pdf

The first reason is modal decomposition (use $x=Vz$, in which $V$ is the matrix containing the eigenvectors of the matrix $A$). Assuming we have n distinct eigenvalues, then modal decomposition helps us to turn a difficult system into a diagonal system, in which we can solve each line of the ODE by a simple integration.