Modal matrix


In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.
Specifically the modal matrix for the matrix is the n × n matrix formed with the eigenvectors of as columns in. It is utilized in the similarity transformation
where is an n × n diagonal matrix with the eigenvalues of on the main diagonal of and zeros elsewhere. The matrix 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.

Example

The matrix
has eigenvalues and corresponding eigenvectors
A diagonal matrix, similar to is
One possible choice for an invertible matrix such that is
Note that since eigenvectors themselves are not unique, and since the columns of both and may be interchanged, it follows that both and are not unique.

Generalized modal matrix

Let be an n × n matrix. A generalized modal matrix for is an n × n matrix whose columns, considered as vectors, form a canonical basis for and appear in according to the following rules:
One can show that
where is a matrix in Jordan normal form. By premultiplying by, we obtain
Note that when computing these matrices, equation is the easiest of the two equations to verify, since it does not require inverting a matrix.

Example

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.
The matrix
has a single eigenvalue with algebraic multiplicity. A canonical basis for will consist of one linearly independent generalized eigenvector of rank 3, 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,.
An "almost diagonal" matrix in Jordan normal form, similar to is obtained as follows:
where is a generalized modal matrix for, the columns of are a canonical basis for, and. Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both and may be interchanged, it follows that both and are not unique.