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
the transformed system will be 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
Since each column
where
is a diagonal matrix having the (complex) eigenvalues of
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,
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.