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 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.

Example

The matrix has eigenvalues and corresponding eigenvectors A diagonal matrix D, similar to A is One possible choice for an invertible matrix M such that is 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.

Generalized modal matrix

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: 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 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 \mu_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, one chain of two vectors , and two chains of one vector ,. An "almost diagonal" matrix J in Jordan normal form, similar to A is obtained as follows: where M is a generalized modal matrix for A, the columns of M are a canonical basis for A, and AM = MJ. 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.