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:
“As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”
—Margaret Atwood (b. 1939)