Matrix Diagonalization

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

Similarity transformation has several properties:

• Identity : a matrix is similar to itself
• Commutative : if is similar to then is similar to .
• Transitive : if is similar to and is similar to , then is similar to .
• If is similar to then they have
• Equal determinant : • Equal rank : • Equal trace : • Equal transpose : • Equal inverse : • Equal matrix power : • Equal matrix exponent : 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 such that . We can form modal matrix from the eigenvector of matrix . However, we are not sure if the modal matrix is nonsingular (has inverse).

We know that modal matrix is nonsingular when the eigenvectors of the square matrix are being linearly independent . But, again we are not sure whether the eigenvectors of the square matrix will be linearly independent. We only know that if all the eigenvalues of the square matrix 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 .

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

Example:

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

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

Example:

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