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Modal matrix

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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

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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

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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

(1)

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

(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.[6]

Example

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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

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

References

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  • 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
  • Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646