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

where may be complex. In other words, when the matrix of the similarity transformation is composed of the eigenvectors of ,

the transformed system will be

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

Since each column of is an eigenvector of , we have , , which implies

where

is a diagonal matrix having the (complex) eigenvalues of along the diagonal. It then follows that

which shows that the new state transition matrix is diagonal and made up of the eigenvalues of .

The transfer function is now, from Eq.
(G.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.

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