Modal matrix

In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors. Specifically the modal matrix [math]\displaystyle{ M }[/math] for the matrix [math]\displaystyle{ A }[/math] is the n × n matrix formed with the eigenvectors of [math]\displaystyle{ A }[/math] as columns in [math]\displaystyle{ M }[/math]. It is utilized in the similarity transformation

[math]\displaystyle{ D = M^{-1}AM, }[/math]

where [math]\displaystyle{ D }[/math] is an n × n diagonal matrix with the eigenvalues of [math]\displaystyle{ A }[/math] on the main diagonal of [math]\displaystyle{ D }[/math] and zeros elsewhere. The matrix [math]\displaystyle{ D }[/math] is called the spectral matrix for [math]\displaystyle{ A }[/math]. The eigenvalues must appear left to right, top to bottom in the same order as their corresponding eigenvectors are arranged left to right in [math]\displaystyle{ M }[/math].

Example

The matrix

[math]\displaystyle{ A = \begin{pmatrix} 3 & 2 & 0 \\ 2 & 0 & 0 \\ 1 & 0 & 2 \end{pmatrix} }[/math]

has eigenvalues and corresponding eigenvectors

[math]\displaystyle{ \lambda_1 = -1, \quad \, \mathbf b_1 = \left( -3, 6, 1 \right) , }[/math]

[math]\displaystyle{ \lambda_2 = 2, \qquad \mathbf b_2 = \left( 0, 0, 1 \right) , }[/math]

[math]\displaystyle{ \lambda_3 = 4, \qquad \mathbf b_3 = \left( 2, 1, 1 \right) . }[/math]

A diagonal matrix [math]\displaystyle{ D }[/math], similar to [math]\displaystyle{ A }[/math] is

[math]\displaystyle{ D = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{pmatrix}. }[/math]

One possible choice for an invertible matrix [math]\displaystyle{ M }[/math] such that [math]\displaystyle{ D = M^{-1}AM, }[/math] is

[math]\displaystyle{ M = \begin{pmatrix} -3 & 0 & 2 \\ 6 & 0 & 1 \\ 1 & 1 & 1 \end{pmatrix}. }[/math]

Note that since eigenvectors themselves are not unique, and since the columns of both [math]\displaystyle{ M }[/math] and [math]\displaystyle{ D }[/math] may be interchanged, it follows that both [math]\displaystyle{ M }[/math] and [math]\displaystyle{ D }[/math] are not unique.

Generalized modal matrix

Let [math]\displaystyle{ A }[/math] be an n × n matrix. A generalized modal matrix [math]\displaystyle{ M }[/math] for [math]\displaystyle{ A }[/math] is an n × n matrix whose columns, considered as vectors, form a canonical basis for [math]\displaystyle{ A }[/math] and appear in [math]\displaystyle{ M }[/math] according to the following rules:

• All Jordan chains consisting of one vector (that is, one vector in length) appear in the first columns of [math]\displaystyle{ M }[/math].
• All vectors of one chain appear together in adjacent columns of [math]\displaystyle{ M }[/math].
• Each chain appears in [math]\displaystyle{ M }[/math] in order of increasing rank (that is, the generalized eigenvector of rank 1 appears before the generalized eigenvector of rank 2 of the same chain, which appears before the generalized eigenvector of rank 3 of the same chain, etc.).

One can show that

[math]\displaystyle{ AM = MJ, }[/math]

(1)

where [math]\displaystyle{ J }[/math] is a matrix in Jordan normal form. By premultiplying by [math]\displaystyle{ M^{-1} }[/math], we obtain

[math]\displaystyle{ J = M^{-1}AM. }[/math]

(2)

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.

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

[math]\displaystyle{ A = \begin{pmatrix} -1 & 0 & -1 & 1 & 1 & 3 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 2 & 1 & 2 & -1 & -1 & -6 & 0 \\ -2 & 0 & -1 & 2 & 1 & 3 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ -1 & -1 & 0 & 1 & 2 & 4 & 1 \end{pmatrix} }[/math]

has a single eigenvalue [math]\displaystyle{ \lambda_1 = 1 }[/math] with algebraic multiplicity [math]\displaystyle{ \mu_1 = 7 }[/math]. A canonical basis for [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ \left\{ \mathbf x_3, \mathbf x_2, \mathbf x_1 \right\} }[/math], one chain of two vectors [math]\displaystyle{ \left\{ \mathbf y_2, \mathbf y_1 \right\} }[/math], and two chains of one vector [math]\displaystyle{ \left\{ \mathbf z_1 \right\} }[/math], [math]\displaystyle{ \left\{ \mathbf w_1 \right\} }[/math].

An "almost diagonal" matrix [math]\displaystyle{ J }[/math] in Jordan normal form, similar to [math]\displaystyle{ A }[/math] is obtained as follows:

[math]\displaystyle{ M = \begin{pmatrix} \mathbf z_1 & \mathbf w_1 & \mathbf x_1 & \mathbf x_2 & \mathbf x_3 & \mathbf y_1 & \mathbf y_2 \end{pmatrix} = \begin{pmatrix} 0 & 1 & -1 & 0 & 0 & -2 & 1 \\ 0 & 3 & 0 & 0 & 1 & 0 & 0 \\ -1 & 1 & 1 & 1 & 0 & 2 & 0 \\ -2 & 0 & -1 & 0 & 0 & -2 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & -1 & 0 \end{pmatrix}, }[/math]

[math]\displaystyle{ J = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}, }[/math]

where [math]\displaystyle{ M }[/math] is a generalized modal matrix for [math]\displaystyle{ A }[/math], the columns of [math]\displaystyle{ M }[/math] are a canonical basis for [math]\displaystyle{ A }[/math], and [math]\displaystyle{ AM = MJ }[/math]. Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both [math]\displaystyle{ M }[/math] and [math]\displaystyle{ J }[/math] may be interchanged, it follows that both [math]\displaystyle{ M }[/math] and [math]\displaystyle{ J }[/math] are not unique.

References

• Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X, https://archive.org/details/firstcourseinlin0000beau 
• Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press 
• Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: John Wiley & Sons