Matrix Size & Validation
What is Vector?
Vector Scalar Multiple
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
What is a matrix?
Matrix Diagonal Is Diagonal Matrix?
Matrix Basic Operation
Is Equal Matrix?
Matrix Scalar Multiple
Elementary Row Operations
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Is Singular Matrix?
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Nullity & Null Space
Matrix Eigen Value & Eigen Vector
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
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.
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.
Preferable reference for this tutorial is
Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\