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Concept# Modal matrix

Summary

In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.
Specifically the modal matrix for the matrix is the n × n matrix formed with the eigenvectors of as columns in . It is utilized in the similarity transformation
where is an n × n diagonal matrix with the eigenvalues of on the main diagonal of and zeros elsewhere. The matrix is called the spectral matrix for . The eigenvalues must appear left to right, top to bottom in the same order as their corresponding eigenvectors are arranged left to right in .
The matrix
has eigenvalues and corresponding eigenvectors
A diagonal matrix , similar to is
One possible choice for an invertible matrix such that is
Note that since eigenvectors themselves are not unique, and since the columns of both and may be interchanged, it follows that both and are not unique.
Let be an n × n matrix. A generalized modal matrix for is an n × n matrix whose columns, considered as vectors, form a canonical basis for and appear in according to the following rules:
All Jordan chains consisting of one vector (that is, one vector in length) appear in the first columns of .
All vectors of one chain appear together in adjacent columns of .
Each chain appears in 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
where is a matrix in Jordan normal form. By premultiplying by , 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.
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
has a single eigenvalue with algebraic multiplicity .

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In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis. There may not always exist a full set of linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable.

Jordan normal form

In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. Let V be a vector space over a field K.

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