Diagonalizability 2.13 2001/03/23 2002/10/24 Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu Roy Ha rha@rice.edu Elizabeth Chan lizychan@rice.edu Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu eigenvalue eigenvector diagonal Diagonalizability of Matrices A diagonal matrix is one whose elements not on the diagonal are equal to 0. The following matrix is one example. a000 0b00 00c0 000d A matrix A is diagonalizable if there exists a matrix V × n n , V 0 such that V A V Λ is diagonal. In such a case, the diagonal entries of Λ are the eigenvalues of A. Let's take an eigenvalue decomposition example to work backwards to this result. Assume that the matrix A has eigenvectors v and w and the respective eigenvalues λv and λw: A v λv v A w λw w We can combine these two equations into an equation of matrices: A vw vw λv0 0λv To simplify this equation, we can replace the eigenvector matrix with V and the eigenvalue matrix with Λ. A V V Λ Now, by multiplying both sides of the equation by V , we see the diagonalizability equation discussed above. A V Λ V When is such a diagonalization possible? The condition is that the algebraic multiplicity equal the geometric multiplicity for each eigenvalue, αi γi . This makes sense; basically, we are saying that there are as many eigenvectors as there are eigenvalues. If it were not like this, then the V matrices would not be square, and therefore could not be inverted as is required by the diagonalizability equation. Remember that the eigenspace associated with a certain eigenvalue λ is given by ker A λ I . This concept of diagonalizability will come in handy in different linear algebra manipulations later. We can however, see a time-saving application of it now. If the matrix A is diagonalizable, and we know its eigenvalues λi, then we can immediately find the eigenvalues of A 2 : A 2 V Λ V V Λ V V Λ 2 V The eigenvalues of A 2 are simply the eigenvalues of A, squared.