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Matrix Size & Validation
Vector Algebra What is Vector? Vector Norm Unit Vector Vector Addition Vector Subtraction Vector Scalar Multiple Vector Multiplication Vector Inner Product Vector Outer Product Vector Cross Product Vector Triple Cross Product Vector Triple Dot Product Scalar Triple Product Orthogonal & Orthonormal Vector Cos Angle of Vectors Scalar and Vector Projection Matrix Algebra What is a matrix? Special Matrices Matrix One Null Matrix Matrix Diagonal Is Diagonal Matrix? Identity Matrix Matrix Determinant Matrix Sum Matrix Trace Matrix Basic Operation Is Equal Matrix? Matrix Transpose Matrix Addition Matrix Subtraction Matrix Multiplication Matrix Scalar Multiple Hadamard Product Horizontal Concatenation Vertical Concatenation Elementary Row Operations Matrix RREF Finding inverse using RREF (Gauss-Jordan) Finding Matrix Rank using RREF Matrix Inverse Is Singular Matrix? Linear Transformation Matrix Generalized Inverse Solving System of Linear Equations Linear combination, Span & Basis Vector Linearly Dependent & Linearly Independent Change of basis Matrix Rank Matrix Range Matrix Nullity & Null Space Eigen System Matrix Eigen Value & Eigen Vector Symmetric Matrix Matrix Eigen Value & Eigen Vector for Symmetric Matrix Similarity Transformation and Matrix Diagonalization Matrix Power Orthogonal Matrix Spectral Decomposition Singular Value Decomposition Resources on Linear Algebra |
Similarity Transformation and Matrix Diagonalization A square 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 We know that modal matrix To obtain modal matrix Example: Find diagonal matrix of matrix Example:
Example: Matrix
See also: Matrix Eigen Value & Eigen Vector, Matrix Power , Equal matrix Rate this tutorial or give your comments about this tutorial Preferable reference for this tutorial is Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\ |
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is similar to a square matrix 
if there is a non-singular matrix such that
. Let us call matrix 
as a modal matrix.
is similar to 
then 
is similar to
.
is similar to 
and 
is similar to 
, then 
is similar to 
.
is similar to 
then they have
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).
is nonsingular when the eigenvectors of the square matrix 
are being 
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
.
, we perform 
linearly independent 
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
.
. 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.
. The inverse modal matrix is
.
. Notice that the diagonal elements are the eigenvalues. Modal matrix 
is not 
.
has eigenvalues
(with 
(simple). The first eigenvalue 
has corresponding eigenvector 
and 
. The first eigenvalue 
has 
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.
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.