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,

˙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)‖Δij‖‖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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