In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.
Specifically the modal matrix M for the matrix A is the n × n matrix formed with the eigenvectors of A as columns in M . It is utilized in the similarity transformation
D = M − 1 A M , where D is an n × n diagonal matrix with the eigenvalues of A on the main diagonal of D and zeros elsewhere. The matrix D is called the spectral matrix for A . The eigenvalues must appear left to right, top to bottom in the same order as their corresponding eigenvectors are arranged left to right in M .
The matrix
A = ( 3 2 0 2 0 0 1 0 2 ) has eigenvalues and corresponding eigenvectors
λ 1 = − 1 , b 1 = ( − 3 , 6 , 1 ) , λ 2 = 2 , b 2 = ( 0 , 0 , 1 ) , λ 3 = 4 , b 3 = ( 2 , 1 , 1 ) . A diagonal matrix D , similar to A is
D = ( − 1 0 0 0 2 0 0 0 4 ) . One possible choice for an invertible matrix M such that D = M − 1 A M , is
M = ( − 3 0 2 6 0 1 1 1 1 ) . Note that since eigenvectors themselves are not unique, and since the columns of both M and D may be interchanged, it follows that both M and D are not unique.
Let A be an n × n matrix. A generalized modal matrix M for A is an n × n matrix whose columns, considered as vectors, form a canonical basis for A and appear in M according to the following rules:
All Jordan chains consisting of one vector (that is, one vector in length) appear in the first columns of M .All vectors of one chain appear together in adjacent columns of M .Each chain appears in M 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 J is a matrix in Jordan normal form. By premultiplying by M − 1 , we obtain
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.
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
A = ( − 1 0 − 1 1 1 3 0 0 1 0 0 0 0 0 2 1 2 − 1 − 1 − 6 0 − 2 0 − 1 2 1 3 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 − 1 − 1 0 1 2 4 1 ) has a single eigenvalue λ 1 = 1 with algebraic multiplicity μ 1 = 7 . A canonical basis for A 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 { x 3 , x 2 , x 1 } , one chain of two vectors { y 2 , y 1 } , and two chains of one vector { z 1 } , { w 1 } .
An "almost diagonal" matrix J in Jordan normal form, similar to A is obtained as follows:
M = ( z 1 w 1 x 1 x 2 x 3 y 1 y 2 ) = ( 0 1 − 1 0 0 − 2 1 0 3 0 0 1 0 0 − 1 1 1 1 0 2 0 − 2 0 − 1 0 0 − 2 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 − 1 0 − 1 0 ) , J = ( 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 ) , where M is a generalized modal matrix for A , the columns of M are a canonical basis for A , and A M = M J . Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both M and J may be interchanged, it follows that both M and J are not unique.
