Diagonalization
Diagonalization
Given an n x n matrix and let for any column vector in (in standard coordinates). Sometimes we can simplify the matrix representation of drastically if we choose a "nice" coordinate system. For example, if there are n linearly independent eigenvectors of , then they can form a basis for , which is called the eigenbasis of . Using it to define a new coordinate system, the new matrix representation of T will be , where i.e. is an eigenvector corresponding to the eigenvalue for . In other words, , where i.e. the matrix formed by the column vectors of the eigenvectors (in standard coordinates). Hence, is similar to the diagonal matrix and this process is called diagonalization. A matrix is said to be diagonalizable if it is similar to a diagonal matrix. The following theorem gives a characterization of diagonalizable matrices:
Theorem: An n x n matrix is diagonalizable if and only if has n linearly independent eigenvectors.
Example 1: Recall the previous example . We have already known that is an eigenvector of corresponding to and is an eigenvector of corresponding to . Therefore, we define and , where . In other words,
(Note: the matrix in the diagonalization is not unique as there are many choices of eigenvectors.)
Example 2: The matrx is not diagonalizable because its only eigenvalue is and the eigenspace is , which is not enough to form a basis for .
Example 3: Diagonalize , if possible.
Solution: Compute the eigenvalues by solving the characteristic equation:
Hence, the eigenvalues of are .
When , solve . The matrix is row reduced to the matrix in echelon form :
The solutions in the parametric vector form are .
When , solve . The matrix is row reduced to the matrix in echelon form :
The solutions in the parametric vector form are .
We have three eigenvectors that are linearly independent. Therefore, they form an eignebasis for and is diagonalizable:
(Note: The eigenvalues in the diagonal matrix must be in the same order as the eigenvectors in .)
Exercise
Diagonalize , if possible.