Diagonalization of Matrices A diagonal matrix is a square matrix in which all elements outside the diagonal are zero. This means that the matrix can be...
Diagonalization of Matrices
A diagonal matrix is a square matrix in which all elements outside the diagonal are zero. This means that the matrix can be decomposed into a product of elementary diagonal matrices.
Theorem: A matrix A is diagonalizable if and only if it is invertible. Furthermore, if A is diagonalizable, then it is invertible and its diagonal elements are its eigenvalues.
Proof:
Suppose A is diagonalizable. Then A can be expressed as A = PDP^{-1}, where P is a matrix of eigenvectors and D is a diagonal matrix of eigenvalues.
Conversely, suppose that A is invertible and has diagonal elements. Then P^T A P = D, where D is the diagonal matrix of eigenvalues.
Therefore, the theorem holds.
Examples:
A diagonal matrix with diagonal elements 1, 2, 3 is diagonalizable.
A matrix with the following diagonal elements is diagonalizable:
1 0 0
0 2 0
0 0 3
0 0 1
0 0 0
0 0 0
Applications of Diagonalization:
Diagonalization can be used to solve linear systems of equations.
Diagonalization can be used to compute the eigenvalues and eigenvectors of a matrix.
Diagonalization can be used to compute the singular value decomposition of a matrix