Modal matrix

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

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.

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

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

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 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.).

One can show that

$AM=MJ,$ 1

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

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

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