In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.
[1] Specifically the modal matrix
is the n × n matrix formed with the eigenvectors of
It is utilized in the similarity transformation where
is an n × n diagonal matrix with the eigenvalues of
on the main diagonal of
is called the spectral matrix for
The eigenvalues must appear left to right, top to bottom in the same order as their corresponding eigenvectors are arranged left to right in
[2] The matrix has eigenvalues and corresponding eigenvectors A diagonal matrix
is One possible choice for an invertible matrix
is Note that since eigenvectors themselves are not unique, and since the columns of both
A generalized modal matrix
is an n × n matrix whose columns, considered as vectors, form a canonical basis for
according to the following rules: One can show that where
is a matrix in Jordan normal form.
, we obtain Note that when computing these matrices, equation (1) is the easiest of the two equations to verify, since it does not require inverting a matrix.
[6] This example illustrates a generalized modal matrix with four Jordan chains.
Unfortunately, it is a little difficult to construct an interesting example of low order.
[7] The matrix has a single eigenvalue
with algebraic multiplicity
A canonical basis for
will consist of one linearly independent generalized eigenvector of rank 3 (generalized eigenvector rank; see generalized eigenvector), two of rank 2 and four of rank 1; or equivalently, one chain of three vectors
, one chain of two vectors
, and two chains of one vector
in Jordan normal form, similar to
is a generalized modal matrix for
[8] Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both