### Diagonalizing a State-Space Model

To obtain the*modal representation*, we may

*diagonalize*any state-space representation. This is accomplished by means of a particular

*similarity transformation*specified by the

*eigenvectors*of the state transition matrix . An

*eigenvector*of the square matrix is any vector for which

*diagonalized*, as we will see below. A system can be diagonalized whenever the eigenvectors of are

*linearly independent*. This always holds when the system poles are

*distinct*. It may or may not hold when poles are

*repeated*. To see how this works, suppose we are able to find linearly independent eigenvectors of , denoted , . Then we can form an matrix having these eigenvectors as columns. Since the eigenvectors are linearly independent, is full rank and can be used as a one-to-one linear transformation, or

*change-of-coordinates*matrix. From Eq.(G.19), we have that the transformed state transition matrix is given by

We have incidentally shown that the eigenvalues of the state-transition matrix are the poles of the system transfer function. When it is

*diagonal*,

*i.e.*, when diag, the state-space model may be called a

*modal representation*of the system, because the poles appear explicitly along the diagonal of and the system's dynamic modes are decoupled. Notice that the diagonalized state-space form is essentially equivalent to a

*partial-fraction expansion*form (§6.8). In particular, the

*residue*of the th pole is given by . When complex-conjugate poles are combined to form real, second-order blocks (in which case is block-diagonal with blocks along the diagonal), this is corresponds to a partial-fraction expansion into real, second-order, parallel filter sections.

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Matlab State-Space Filter Conversion Example