modal matrix

In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.{{harvtxt|Bronson|1970|pp=179–183}}

Specifically the modal matrix $M$ for the matrix $A$ is the n × n matrix formed with the eigenvectors of $A$ as columns in $M$ . It is utilized in the similarity transformation

: $D = M^{-1}AM,$

where $D$ is an n × n diagonal matrix with the eigenvalues of $A$ on the main diagonal of $D$ and zeros elsewhere. The matrix $D$ is called the spectral matrix for $A$ . The eigenvalues must appear left to right, top to bottom in the same order as their corresponding eigenvectors are arranged left to right in $M$ .{{harvtxt|Bronson|1970|p=181}}

Example

The matrix

: $A = \begin{pmatrix}
3 & 2 & 0 \\
2 & 0 & 0 \\
1 & 0 & 2
\end{pmatrix}$

has eigenvalues and corresponding eigenvectors

: $\lambda_1 = -1, \quad \, \mathbf b_1 = \left( -3, 6, 1 \right) ,$

: $\lambda_2 = 2, \qquad \mathbf b_2 = \left( 0, 0, 1 \right) ,$

: $\lambda_3 = 4, \qquad \mathbf b_3 = \left( 2, 1, 1 \right) .$

A diagonal matrix $D$ , similar to $A$ is

: $D = \begin{pmatrix}
-1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 4
\end{pmatrix}.$

One possible choice for an invertible matrix $M$ such that $D = M^{-1}AM,$ is

: $M = \begin{pmatrix}
-3 & 0 & 2 \\
6 & 0 & 1 \\
1 & 1 & 1
\end{pmatrix}.$ {{harvtxt|Beauregard|Fraleigh|1973|pp=271,272}}

Note that since eigenvectors themselves are not unique, and since the columns of both $M$ and $D$ may be interchanged, it follows that both $M$ and $D$ are not unique.{{harvtxt|Bronson|1970|p=181}}

Generalized modal matrix

Let $A$ be an n × n matrix. A generalized modal matrix $M$ for $A$ is an n × n matrix whose columns, considered as vectors, form a canonical basis for $A$ and appear in $M$ according to the following rules:

All Jordan chains consisting of one vector (that is, one vector in length) appear in the first columns of $M$ .
All vectors of one chain appear together in adjacent columns of $M$ .
Each chain appears in $M$ 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.).{{harvtxt|Bronson|1970|p=205}}

One can show that

{{NumBlk|:::| $AM = MJ,$ |{{EquationRef|1}}}}

where $J$ is a matrix in Jordan normal form. By premultiplying by $M^{-1}$ , we obtain

{{NumBlk|:::| $J = M^{-1}AM.$ |{{EquationRef|2}}}}

Note that when computing these matrices, equation ({{EquationNote|1}}) is the easiest of the two equations to verify, since it does not require inverting a matrix.{{harvtxt|Bronson|1970|pp=206–207}}

= 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.{{harvtxt|Nering|1970|pp=122,123}}

The matrix

: $A = \begin{pmatrix}
-1 & 0 & -1 & 1 & 1 & 3 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
2 & 1 & 2 & -1 & -1 & -6 & 0 \\
-2 & 0 & -1 & 2 & 1 & 3 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
-1 & -1 & 0 & 1 & 2 & 4 & 1
\end{pmatrix}$

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

An "almost diagonal" matrix $J$ in Jordan normal form, similar to $A$ is obtained as follows:

: $M =
\begin{pmatrix} \mathbf z_1 & \mathbf w_1 & \mathbf x_1 & \mathbf x_2 & \mathbf x_3 & \mathbf y_1 & \mathbf y_2 \end{pmatrix} =
\begin{pmatrix}
0 & 1 & -1 & 0 & 0 & -2 & 1 \\
0 & 3 & 0 & 0 & 1 & 0 & 0 \\
-1 & 1 & 1 & 1 & 0 & 2 & 0 \\
-2 & 0 & -1 & 0 & 0 & -2 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & -1 & 0
\end{pmatrix},$

: $J = \begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{pmatrix},$

where $M$ is a generalized modal matrix for $A$ , the columns of $M$ are a canonical basis for $A$ , and $AM = MJ$ .{{harvtxt|Bronson|1970|pp=208,209}} Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both $M$ and $J$ may be interchanged, it follows that both $M$ and $J$ are not unique.{{harvtxt|Bronson|1970|p=206}}

Notes

References

{{citation | first1 = Raymond A. | last1 = Beauregard | first2 = John B. | last2 = Fraleigh | year = 1973 | isbn = 0-395-14017-X | title = A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields | publisher = Houghton Mifflin Co. | location = Boston | url-access = registration | url = https://archive.org/details/firstcourseinlin0000beau }}
{{ citation | first1 = Richard | last1 = Bronson | year = 1970 | lccn = 70097490 | title = Matrix Methods: An Introduction | publisher = Academic Press | location = New York }}
{{ citation | first1 = Evar D. | last1 = Nering | year = 1970 | title = Linear Algebra and Matrix Theory | edition = 2nd | publisher = Wiley | location = New York | lccn = 76091646 }}

Category:Matrices (mathematics)