Codex Regius (New Testament): Difference between revisions
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In [[linear algebra]], the '''modal matrix''' is used in the [[Diagonalizable|diagonalization process]] involving [[eigenvalues and eigenvectors]]. | |||
Assume a linear system of the following form: | |||
: <math>{d \over dt} X = A^* X + B^* U</math> | |||
where ''X'' is ''n''×1, ''A'' is ''n''×''n'', and ''B'' is ''n''×1. ''X'' typically represents the state vector, and ''U'' the system input. | |||
Specifically the modal matrix ''M'' is the ''n''×''n'' matrix formed with the eigenvectors of ''A'' as columns in ''M''. It is utilized in | |||
: <math> M^{-1}AM = D \, </math> | |||
where ''D'' is an ''n''×''n'' [[diagonal matrix]] with the eigenvalues of ''A'' on the main diagonal of ''D'' and zeros elsewhere. (note the eigenvalues should appear left→right top→bottom in the same order as its eigenvectors are arranged left→right into ''M'') | |||
Note that the modal matrix ''M'' provides the conjugation to make ''A'' and ''D'' [[similar matrices]]. | |||
[[Category:Matrices]] | |||
Revision as of 17:23, 16 March 2013
In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.
Assume a linear system of the following form:
where X is n×1, A is n×n, and B is n×1. X typically represents the state vector, and U the system input.
Specifically the modal matrix M is the n×n matrix formed with the eigenvectors of A as columns in M. It is utilized in
where D is an n×n diagonal matrix with the eigenvalues of A on the main diagonal of D and zeros elsewhere. (note the eigenvalues should appear left→right top→bottom in the same order as its eigenvectors are arranged left→right into M)
Note that the modal matrix M provides the conjugation to make A and D similar matrices.