Modal matrix

The modal matrix is used in linear algebra as well as linear systems analysis in the diagonalization process involving eigenvalues and eigenvectors.

Assume a linear system of the following form:

d/dt(X) = A*X + B*U

where X is nx1, A is nxn, and B is nx1. X typically represents the state vector, and U the system input.

specifically the Modal Matrix M is the nxn matrix formed with A's eigenvectors as columns in M. It is utilized in

(M^-1)(A)(M) = D

where D is an nxn 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)

this process is also known as the similarity transform