In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.
Specifically the modal matrix M {\displaystyle M} for the matrix A {\displaystyle A} is the n × n matrix formed with the eigenvectors of A {\displaystyle A} as columns in M {\displaystyle M} . It is utilized in the similarity transformation
where D {\displaystyle D} is an n × n diagonal matrix with the eigenvalues of A {\displaystyle A} on the main diagonal of D {\displaystyle D} and zeros elsewhere. The matrix D {\displaystyle D} is called the spectral matrix for A {\displaystyle 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 {\displaystyle M} .