Modal matrix

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

Assume a linear system of the following form:

${d \over dt}X=A^{*}X+B^{*}U$

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 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

$M^{-1}AM=D\,$

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.

References

• Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490