

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 (GaussJordan) 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 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 matrix, can we find a scalar number and a vector such that? Any solution of equation for is called eigenvector of. The scalar is called the eigenvalue of matrix. 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 vector. Notice the equation said that multiplication of a matrix by a vector is equal to multiplication of a scalar by the same vector. Thus, the scalar characterizes the matrix. Since eigenvalue is the scalar multiple to eigenvector, geometrically, eigenvalue indicates how much the eigenvectoris shortened or lengthened after multiplication by the matrixwithout changing the vector orientation. Algebraically, we can solve the equation by rearranging it into a homogeneous linear system where matrix is the identity matrix order. A homogeneous linear system has non trivia solution if the matrix is singular. That happens when the determinant is equal to zero, that is. Equation is called the characteristic equation of matrix. Expanding the determinant formula (using cofactor), we will get the solution in the polynomial form with coefficients. This polynomial equation is called the characteristic polynomial of matrix. The solution of the characteristic polynomial of are eigenvalues, 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 equation. We do that for each of the eigenvalue. If is 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 one.
Example: Thus, eigenvalue has corresponding eigenvector and eigenvalue has corresponding eigenvector. 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. PropertiesSome important properties of eigenvalue, eigenvectors and characteristic equation are:
See also: Matrix Eigen Value & Eigen Vector for Symmetric Matrix, Similarity and Matrix Diagonalization, Matrix Power 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\ 



