Generalized Modal Matrix
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linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, the modal matrix is used in the diagonalization process involving
eigenvalues and eigenvectors In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
. 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^AM, where D is an ''n'' × ''n''
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagon ...
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 3 & 2 & 0 \\ 2 & 0 & 0 \\ 1 & 0 & 2 \end 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 -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end. One possible choice for an
invertible matrix In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
M such that D = M^AM, is :M = \begin -3 & 0 & 2 \\ 6 & 0 & 1 \\ 1 & 1 & 1 \end. 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.


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 In linear algebra, a generalized eigenvector of an n\times n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let V be an n-dimensional vector space and let A be the matrix r ...
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 where J is a matrix in
Jordan normal form \begin \lambda_1 1\hphantom\hphantom\\ \hphantom\lambda_1 1\hphantom\\ \hphantom\lambda_1\hphantom\\ \hphantom\lambda_2 1\hphantom\hphantom\\ \hphantom\hphantom\lambda_2\hphantom\\ \hphantom\lambda_3\hphantom\\ \hphantom\ddots\hphantom\\ ...
. By premultiplying by M^ , 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 :A = \begin -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 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 In linear algebra, a generalized eigenvector of an n\times n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let V be an n-dimensional vector space and let A be the matrix r ...
), two of rank 2 and four of rank 1; or equivalently, one chain of three vectors \left\ , one chain of two vectors \left\ , and two chains of one vector \left\ , \left\ . An "almost diagonal" matrix J in ''Jordan normal form'', similar to A is obtained as follows: : M = \begin \mathbf z_1 & \mathbf w_1 & \mathbf x_1 & \mathbf x_2 & \mathbf x_3 & \mathbf y_1 & \mathbf y_2 \end = \begin 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, :J = \begin 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, where M is a generalized modal matrix for A, the columns of M are a canonical basis for A, and AM = MJ. 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.


Notes


References

* * * {{ citation , first1 = Evar D. , last1 = Nering , year = 1970 , title = Linear Algebra and Matrix Theory , edition = 2nd , publisher = Wiley , location = New York , lccn = 76091646 Matrices (mathematics)