Modal matrix

In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.^[1]

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

${\displaystyle D=M^{-1}AM,}$

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 $M$.^[2]

Example

The matrix

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

has eigenvalues and corresponding eigenvectors

${\displaystyle \lambda _{1}=-1,\quad \,{\mathbf {b}}_{1}=\left(-3,6,1\right),}$

${\displaystyle \lambda _{2}=2,\qquad {\mathbf {b}}_{2}=\left(0,0,1\right),}$

${\displaystyle \lambda _{3}=4,\qquad {\mathbf {b}}_{3}=\left(2,1,1\right).}$

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

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

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

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

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.^[4]

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.).^[5]

One can show that

${\displaystyle AM=MJ,}$

(1)

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

${\displaystyle J=M^{-1}AM.}$

(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.^[6]

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.^[7] The matrix

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

has a single eigenvalue $\lambda _{1}=1$ with algebraic multiplicity ${\displaystyle \mu _{1}=7}$. 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 ${\displaystyle \left\{{\mathbf {x}}_{3},{\mathbf {x}}_{2},{\mathbf {x}}_{1}\right\}}$, one chain of two vectors ${\displaystyle \left\{{\mathbf {y}}_{2},{\mathbf {y}}_{1}\right\}}$, and two chains of one vector ${\displaystyle \left\{{\mathbf {z}}_{1}\right\}}$, ${\displaystyle \left\{{\mathbf {w}}_{1}\right\}}$.

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

${\displaystyle M={\begin{pmatrix}{\mathbf {z}}_{1}&{\mathbf {w}}_{1}&{\mathbf {x}}_{1}&{\mathbf {x}}_{2}&{\mathbf {x}}_{3}&{\mathbf {y}}_{1}&{\mathbf {y}}_{2}\end{pmatrix}}={\begin{pmatrix}0&1&-1&0&0&-2&1\\0&3&0&0&1&0&0\\-1&1&1&1&0&2&0\\-2&0&-1&0&0&-2&0\\1&0&0&0&0&0&0\\0&1&0&0&0&0&0\\0&0&0&-1&0&-1&0\end{pmatrix}},}$

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

where $M$ is a generalized modal matrix for $A$, the columns of $M$ are a canonical basis for $A$, and $AM=MJ$.^[8] 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.^[9]

Notes

1. Bronson (1970, pp. 179-183)
2. Bronson (1970, p. 181)
3. Beauregard & Fraleigh (1973, pp. 271,272)
4. Bronson (1970, p. 181)
5. Bronson (1970, p. 205)
6. Bronson (1970, pp. 206-207)
7. Nering (1970, pp. 122,123)
8. Bronson (1970, pp. 208,209)
9. Bronson (1970, p. 206)

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
• Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490
• Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646