By Kardi Teknomo, PhD .
LinearAlgebra

< Next | Previous | Index >

Matrix Diagonalization

A square matrix Similarity Transformation and Matrix Diagonalization is similar to a square matrix Similarity Transformation and Matrix Diagonalization if there is a non-singular matrix such that Similarity Transformation and Matrix Diagonalization . Let us call matrix Similarity Transformation and Matrix Diagonalization as a modal matrix.

Similarity transformation has several properties:

Since diagonal matrix has many nice properties similar to a scalar, we would like to find matrix similarity to a diagonal matrix. The only requirement to perform similarity transformation is to find a non singular modal matrix Similarity Transformation and Matrix Diagonalization such that Similarity Transformation and Matrix Diagonalization . We can form modal matrix Similarity Transformation and Matrix Diagonalization from the eigenvector of matrix Similarity Transformation and Matrix Diagonalization . However, we are not sure if the modal matrix Similarity Transformation and Matrix Diagonalization is nonsingular (has inverse).

We know that modal matrix Similarity Transformation and Matrix Diagonalization is nonsingular when the eigenvectors of the square matrix Similarity Transformation and Matrix Diagonalization are being linearly independent . But, again we are not sure whether the eigenvectors of the square matrix Similarity Transformation and Matrix Diagonalization will be linearly independent. We only know that if all the eigenvalues of the square matrix Similarity Transformation and Matrix Diagonalization are distinct (do not have any eigenvalue of multiple values) then the eigenvectors are linearly independent. Thus, any square matrix with distinct eigenvalues can be converted into diagonal matrix by similarity transformation Similarity Transformation and Matrix Diagonalization .

To obtain modal matrix Similarity Transformation and Matrix Diagonalization , we perform horizontal concatenation of the Similarity Transformation and Matrix Diagonalization linearly independent eigenvectors of matrix Similarity Transformation and Matrix Diagonalization such that Similarity Transformation and Matrix Diagonalization . Since eigenvalues of matrix Similarity Transformation and Matrix Diagonalization are all distinct, modal matrix Similarity Transformation and Matrix Diagonalization has full rank because the eigenvectors are linearly independent, therefore modal matrix Similarity Transformation and Matrix Diagonalization has inverse (nonsingular). The diagonal elements of diagonal matrix Similarity Transformation and Matrix Diagonalization consist of the eigenvalues of Similarity Transformation and Matrix Diagonalization .

Example:

Find diagonal matrix of matrix Similarity Transformation and Matrix Diagonalization
Solution: First, we find the eigenvector and eigenvalues of matrix Similarity Transformation and Matrix Diagonalization . The matrix has 2 distinct eigenvalues. Eigenvalue Similarity Transformation and Matrix Diagonalization has corresponding eigenvector Similarity Transformation and Matrix Diagonalization and eigenvalue Similarity Transformation and Matrix Diagonalization has corresponding eigenvector Similarity Transformation and Matrix Diagonalization . Since the eigenvalues are all distinct, the matrix is diagonalizable and the eigenvectors are linearly independent.
Next, we form modal matrix Similarity Transformation and Matrix Diagonalization . The inverse modal matrix is Similarity Transformation and Matrix Diagonalization .
Then, we obtain the diagonal matrix Similarity Transformation and Matrix Diagonalization . Notice that the diagonal elements are the eigenvalues. Modal matrix Similarity Transformation and Matrix Diagonalization is not orthogonal matrix because Similarity Transformation and Matrix Diagonalization .


Example:
Matrix Similarity Transformation and Matrix Diagonalization has eigenvalues Similarity Transformation and Matrix Diagonalization (with algebraic multiplicity of 2) and Similarity Transformation and Matrix Diagonalization (simple). The first eigenvalue Similarity Transformation and Matrix Diagonalization has corresponding eigenvector Similarity Transformation and Matrix Diagonalization and Similarity Transformation and Matrix Diagonalization . The first eigenvalue Similarity Transformation and Matrix Diagonalization has geometric multiplicity of 2 because the two eigenvectors Similarity Transformation and Matrix Diagonalization and Similarity Transformation and Matrix Diagonalization are linearly independent. The second eigenvalue Similarity Transformation and Matrix Diagonalization has corresponding eigenvector Similarity Transformation and Matrix Diagonalization . Since matrix Similarity Transformation and Matrix Diagonalization has 3 linearly independent eigenvectors, matrix Similarity Transformation and Matrix Diagonalization is non-defective. We can form modal matrix Similarity Transformation and Matrix Diagonalization such that Similarity Transformation and Matrix Diagonalization . Notice in this example that the eigenvalues are not all distinct but the eigenvectors are linearly independent, therefore the matrix is diagonalizable.

Example:

Matrix Non-Diagonalizable has multiple eigenvalue of Non-Diagonalizable . The eigenvectors associated with eigenvalues are vectors of the form of Non-Diagonalizable for Non-Diagonalizable any non-zero real number. Since the eigenvectors are linearly dependent, the modal matrix Non-Diagonalizable has no inverse and therefore matrix Non-Diagonalizable is non-diagonalizable.

See also : Matrix Eigen Value & Eigen Vector , Matrix Power , Equal matrix

Rate this tutorial or give your comments about this tutorial

< Next | Previous | Index >

This tutorial is copyrighted .