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

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. (E.19), we have that
the transformed state transition matrix is given by

The transfer function is now, from Eq. (E.5), in the SISO case,

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

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.

[How to cite this work] [Order a printed hardcopy]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

[Automatic-links disclaimer]