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,

\[\dot{x}=Ax\]

The laplace transform of the governing equations is as follows,

\[sx(s)-x(0)=Ax(s)\]

Which essentially means the following,

\[x(t)=L^{-1}((sI-A)^{-1})x(0)\]

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,

\[-1^{(i+j)}\frac{\|\Delta_{ij}\|}{\|sI-A\|}\]

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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