Kardi Teknomo
Kardi Teknomo Kardi Teknomo Kardi Teknomo
   
 
  Research
  Publications
  Tutorials
  Resume
  Personal
  Contact

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

Eigen Value and Eigen Vector

By Kardi Teknomo, PhD.
LinearAlgebra

<Next | Previous | Index>

A matrix usually consists of many scalar elements. Can we characterize a matrix by a few numbers? In particular, our question is given a square matrixEigen Value and Eigen Vector, can we find a scalar number Eigen Value and Eigen Vectorand a vector Eigen Value and Eigen Vectorsuch thatEigen Value and Eigen Vector? Any solution of equation Eigen Value and Eigen Vectorfor Eigen Value and Eigen Vector is called eigenvector ofEigen Value and Eigen Vector. The scalar is called the eigenvalue of matrixEigen Value and Eigen Vector.

Eigenvalue is also called proper value, characteristic value, latent value, or latent root. Similarly, eigenvector is also called proper vector, characteristic vector, or latent vector.

In the topic of Linear Transformation, we learned that a multiplication of a matrix with a vector will produce the transformation of the vectorEigen Value and Eigen Vector. Notice the equation Eigen Value and Eigen Vectorsaid that multiplication of a matrix by a vector is equal to multiplication of a scalar by the same vector. Thus, the scalar Eigen Value and Eigen Vectorcharacterizes the matrixEigen Value and Eigen Vector.

Since eigenvalue Eigen Value and Eigen Vectoris the scalar multiple to eigenvectorEigen Value and Eigen Vector, geometrically, eigenvalue indicates how much the eigenvectorEigen Value and Eigen Vectoris shortened or lengthened after multiplication by the matrixEigen Value and Eigen VectorEigen Value and Eigen Vectorwithout changing the vector orientation.

Algebraically, we can solve the equation Eigen Value and Eigen Vectorby rearranging it into a homogeneous linear system Eigen Value and Eigen Vectorwhere matrix Eigen Value and Eigen Vectoris the identity matrix orderEigen Value and Eigen Vector. A homogeneous linear system has non trivia solution if the matrix Eigen Value and Eigen Vectoris singular. That happens when the determinant is equal to zero, that isEigen Value and Eigen Vector. Equation is called the characteristic equation of matrixEigen Value and Eigen Vector.

Expanding the determinant formula (using cofactor), we will get the solution in the polynomial form with coefficientsEigen Value and Eigen Vector. This polynomial equation Eigen Value and Eigen Vectoris called the characteristic polynomial of matrixEigen Value and Eigen Vector. The solution of the characteristic polynomial of Eigen Value and Eigen Vectorare Eigen Value and Eigen Vectoreigenvalues, some eigenvalues may be identical (the same eigenvalues) and some eigenvalues may be complex numbers.

Each eigenvalue has a corresponding eigenvector. To find the eigenvector, we put back the eigenvalue into equationEigen Value and Eigen Vector. We do that for each of the eigenvalue. If Eigen Value and Eigen Vectoris an eigenvector of A, then any scalar multiple is also an eigenvector with the same eigenvalue. We often use normalized eigenvector into unit vector such that the inner product with itself is oneEigen Value and Eigen Vector.

 

Example:
Find eigenvalues and eigenvectors of matrixEigen Value and Eigen Vector
Solution: we form characteristic equationEigen Value and Eigen Vector
Eigen Value and Eigen Vector
The eigenvalues areEigen Value and Eigen Vector andEigen Value and Eigen Vector.
For the first eigenvalueEigen Value and Eigen Vector, the system equation isEigen Value and Eigen Vector
Eigen Value and Eigen Vector
Eigen Value and Eigen Vector
The two rows are equivalent and produces equationEigen Value and Eigen Vector. This is an equation of a line with many solutions, we can put arbitrary value Eigen Value and Eigen Vectorto obtain
Eigen Value and Eigen Vector. You can also write as Eigen Value and Eigen Vector or Eigen Value and Eigen Vector and they lie on the same line.
The normalized eigenvector is Eigen Value and Eigen Vector
For the second eigenvalueEigen Value and Eigen Vector, the eigenvector is computed from the system equation Eigen Value and Eigen Vector
Eigen Value and Eigen Vector
Eigen Value and Eigen Vector
The two rows are equivalent and produces equationEigen Value and Eigen Vector. This is an equation of a line with many solutions, arbitrarily we can put Eigen Value and Eigen Vectorto obtain
Eigen Value and Eigen Vector. You can also write as Eigen Value and Eigen Vector or Eigen Value and Eigen Vector and they lie on the same line.
The normalized eigenvector isEigen Value and Eigen Vector

Thus, eigenvalue Eigen Value and Eigen Vectorhas corresponding eigenvectorEigen Value and Eigen Vector and eigenvalue Eigen Value and Eigen Vectorhas corresponding eigenvectorEigen Value and Eigen Vector.  Note that the eigenvectors are actually lines with many solutions and we put only one of the solutions. They are correct up to a scalar multiple. Since the eigenvalues are all distinct, the matrix is diagonalizable and the eigenvectors are linearly independent.

Properties

Some important properties of eigenvalue, eigenvectors and characteristic equation are:

  • Every square matrix has at least one eigenvalue a corresponding non zero eigenvector.
  • When a square matrix has multiple eigenvalues (that is repeated, non-distinct eigenvalues), we have two terms to characterize the complexity of the matrix:
    • The algebraic multiplicity of an eigenvalueEigen Value and Eigen Vector is the integer Eigen Value and Eigen Vectorassociated with Eigen Value and Eigen Vectorwhen it appears in the characteristic polynomial. If the algebraic multiplicity is one, the eigenvalues is said to be simple.
    • The geometric multiplicity of Eigen Value and Eigen Vectoris the number of linearly independent eigenvectors that can be associated withEigen Value and Eigen Vector. For any eigenvalue, the geometric multiplicity is always at least one. Geometric multiplicity never exceeds algebraic multiplicity.
    Example:
    Matrix Eigen Value and Eigen Vectorhas characteristic polynomial Eigen Value and Eigen Vectorthus the eigenvalue is 6 with algebraic multiplicity of 2. There is only one linearly independent eigenvectorEigen Value and Eigen Vector, thus the geometric multiplicity is 1.
  • The eigenvectors that belong to distinct eigenvalues are linearly independent eigenvectors. This is true even if the eigenvalues are not all distinct.
  • If a square matrix Eigen Value and Eigen Vectorhas fewer than Eigen Value and Eigen Vectorlinearly independent eigenvector, then matrix Eigen Value and Eigen Vectoris called defective matrix. Defective matrix is not diagonalizable.
  • When the eigenvalues of a square matrix Eigen Value and Eigen Vectorare all distinct (no multiple eigenvalues), we called it non-defective matrix. A non-defective matrix has Eigen Value and Eigen Vectorlinearly independent eigenvectors that can form a basis (i.e. a coordinate system) for Eigen Value and Eigen Vectordimensional space. Non-defective matrixEigen Value and Eigen Vector is diagonalizable by similarity transformation Eigen Value and Eigen Vectorinto a diagonal matrix Eigen Value and Eigen Vector. The eigenvalues of Eigen Value and Eigen Vector lie on the main diagonal of Eigen Value and Eigen Vector. Modal matrix Eigen Value and Eigen Vector is formed by horizontal concatenation of the Eigen Value and Eigen Vectorlinearly independent eigenvectorsEigen Value and Eigen Vector.
  • If matrixEigen Value and Eigen Vectoris symmetric then matrix Eigen Value and Eigen Vectorhas linearly independent Eigen vectors and the Eigen values of symmetric matrix Eigen Value and Eigen Vectorare all real numbers (no complex numbers).
  • If all eigenvalues of symmetric matrix Eigen Value and Eigen Vectorare distinct (all eigenvalues are simple), then matrix Eigen Value and Eigen Vector can be transformed into a diagonal matrix. Furthermore, the eigenvectors are orthogonal.
  • Matrix Eigen Value and Eigen Vector satisfies its own characteristic equation. If polynomial Eigen Value and Eigen Vectoris the characteristics polynomial equation of a square matrix A, then matrix Eigen Value and Eigen Vector satisfies Cayley-Hamilton equationEigen Value and Eigen Vector.

See also: Matrix Eigen Value & Eigen Vector for Symmetric Matrix, Similarity and Matrix Diagonalization, Matrix Power

<Next | Previous | Index>

Rate this tutorial or give your comments about this tutorial

This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\

 

 
© 2007 Kardi Teknomo. All Rights Reserved.
Designed by CNV Media