Eigen Value and Eigen Vector (original) (raw)

By Kardi Teknomo, PhD.
LinearAlgebra

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Eigenvalue and Eigenvector

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:

Interactive Eigenvalue and eigen vectors below are useful to compute general square matrix. You can select from the matrix example and click 'Compute Eigenvalues and Eigenvectors' button. This is a working progress, thus the result are not perfect yet, sometimes it would still produce inaccurate results.

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

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Preferable reference for this tutorial is

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