“Modal matrix”的意思、由来-开放百科全书

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Modal matrix

释义

Example
Generalized modal matrix
Example
Notes
References

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

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 .^[2]

Example

The matrix

has eigenvalues and corresponding eigenvectors

A diagonal matrix , similar to is

One possible choice for an invertible matrix such that is

^[3]

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.^[4]

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 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.).^[5]

One can show that

{{NumBlk|:||{{EquationRef|1}}}}

where is a matrix in Jordan normal form. By premultiplying by , we obtain

{{NumBlk|:||{{EquationRef|2}}}}

Note that when computing these matrices, equation ({{EquationNote|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

has a single eigenvalue with algebraic multiplicity . A canonical basis for 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 , 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 .^[8] 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.^[9]

Notes

1. ^{{harvtxt|Bronson|1970|pp=179–183}}
2. ^{{harvtxt|Bronson|1970|p=181}}
3. ^{{harvtxt|Beauregard|Fraleigh|1973|pp=271,272}}
4. ^{{harvtxt|Bronson|1970|p=181}}
5. ^{{harvtxt|Bronson|1970|p=205}}
6. ^{{harvtxt|Bronson|1970|pp=206–207}}
7. ^{{harvtxt|Nering|1970|pp=122,123}}
8. ^{{harvtxt|Bronson|1970|pp=208,209}}
9. ^{{harvtxt|Bronson|1970|p=206}}

References

{{ citation | first1 = Raymond A. | last1 = Beauregard | first2 = John B. | last2 = Fraleigh | year = 1973 | isbn = 0-395-14017-X | title = A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields | publisher = Houghton Mifflin Co. | location = Boston }}
{{ citation | first1 = Richard | last1 = Bronson | year = 1970 | lccn = 70097490 | title = Matrix Methods: An Introduction | publisher = Academic Press | location = New York }}
{{ citation | first1 = Evar D. | last1 = Nering | year = 1970 | title = Linear Algebra and Matrix Theory | edition = 2nd | publisher = Wiley | location = New York | lccn = 76091646 }}

1 : Matrices

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