Eigenvalues and poles

As I am used to understanding the linear differential equations using eigenvalues of the coefficient matrix \(A\), I find it very useful to remember the following relationship between this and the poles of the system that control engineers usually talk about.

In other words, this is a note to describe how poles of linear system is related to the eigen values of the coefficient matrix.

Let the governing equations be of the following form,


The laplace transform of the governing equations is as follows,


Which essentially means the following,


Where \((sI-A)^{-1}\) is the resolvant matrix.

Using cramers rule one could compute the inverse of \(sI-A\) matrix. This will result in the \((i,~ j)\) entry of the matrix to be as follows,


Here the poles are governed byt the the term \(\|sI-A\|\) and it is also what governs the eigen values of \(A\). The subte difference comes when the some terms in the denominator gets cancelled by the numerator. This results in a scenario where some eigen values do not appear in the poles of the matrix as they cancel out. In other words all poles are eigen values of the matrix A and all eigen values need not appear as poles as they may get cancelled.


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