In
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 matrices.
...
, the modal matrix is used in the
diagonalization process involving
eigenvalues and eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
.
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
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 diagonal ma ...
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
.
Example
The matrix
:
has eigenvalues and corresponding eigenvectors
:
:
:
A diagonal matrix
,
similar to
is
:
One possible choice for an
invertible matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
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.
Generalized modal matrix
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
In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
* In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the ...
for
and appear in
according to the following rules:
* All
Jordan chain
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; let \phi be a linear map ...
s 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
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; let \phi be a linear map ...
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
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 so ...
. 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.
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
:
has a single eigenvalue
with
algebraic multiplicity
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
. A canonical basis for
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; let \phi be a linear map ...
), two of rank 2 and four of rank 1; or equivalently, one chain of three vectors
, one chain of two vectors
, and two chains of one vector
,
.
An "almost diagonal" matrix
in ''Jordan normal form'', similar to
is obtained as follows:
:
:
where
is a generalized modal matrix for
, the columns of
are a canonical basis for
, and
. Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both
and
may be interchanged, it follows that both
and
are not unique.
Notes
References
*
*
* {{ citation , first1 = Evar D. , last1 = Nering , year = 1970 , title = Linear Algebra and Matrix Theory , edition = 2nd , publisher =
Wiley
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Matrices