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.

### Famous quotes containing the word matrix:

““The *matrix* is God?”

“In a manner of speaking, although it would be more accurate ... to say that the *matrix* has a God, since this being’s omniscience and omnipotence are assumed to be limited to the *matrix*.”

“If it has limits, it isn’t omnipotent.”

“Exactly.... Cyberspace exists, insofar as it can be said to exist, by virtue of human agency.””

—William Gibson (b. 1948)