Define the diagonalization of a matrix.

Diagonalization of a matrix is the process of finding a diagonal matrix that is similar to the given matrix.

To diagonalize a matrix A, we need to find a diagonal matrix D and an invertible matrix P such that A = PDP^-1. The diagonal entries of D are the eigenvalues of A, and the columns of P are the corresponding eigenvectors.

To find the eigenvalues of A, we solve the characteristic equation det(A - λI) = 0, where I is the identity matrix and λ is the eigenvalue. The roots of this equation are the eigenvalues of A.

Once we have found the eigenvalues, we can find the eigenvectors by solving the system of equations (A - λI)x = 0, where x is the eigenvector corresponding to the eigenvalue λ. The eigenvectors must be linearly independent, so we may need to use Gaussian elimination to find a basis for the eigenspace.

Once we have found the eigenvectors, we can form the matrix P by placing the eigenvectors as columns. The matrix D is formed by placing the eigenvalues on the diagonal and zeros elsewhere. Then we have A = PDP^-1, which is the diagonalization of A.

Diagonalization is useful for many applications, such as finding powers of a matrix, computing matrix exponentials, and solving systems of differential equations. It also provides insight into the structure of a matrix and its relationship to its eigenvectors and eigenvalues.