Matrix Diagonalization
A square matrix is similar to a square matrix if there is a nonsingular 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 nondefective. 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 nonzero real number. Since the eigenvectors are linearly dependent, the modal matrix has no inverse and therefore matrix is nondiagonalizable.
See also : Matrix Eigen Value & Eigen Vector , Matrix Power , Equal matrix
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