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 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 of is an eigenvector of , we have , , which implies
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,
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.