3.1 Eigenvalues and eigenvectors of a linear operator
3 min read•Last Updated on August 16, 2024
Eigenvalues and eigenvectors are key concepts in linear algebra. They help us understand how linear transformations affect vector spaces, revealing the fundamental properties of these operations.
These tools are crucial for solving complex problems in various fields. From physics to computer science, eigenvalues and eigenvectors provide insights into system behavior, stability, and optimization techniques.
Eigenvalues and Eigenvectors
Definitions and Properties
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Eigenvalue represents a scalar λ where T(v) = λv for a non-zero vector v in the vector space
Eigenvector constitutes a non-zero vector v satisfying T(v) = λv for a given eigenvalue λ
Eigenspace encompasses all eigenvectors corresponding to a particular eigenvalue λ along with the zero vector
Linear operators possess intrinsic eigenvalues and eigenvectors regardless of chosen basis
Algebraic multiplicity denotes the multiplicity of an eigenvalue as a root of the characteristic polynomial
Geometric multiplicity signifies the dimension of the corresponding eigenspace for an eigenvalue
Multiplicities and Complex Eigenvalues
Algebraic and geometric multiplicities provide comprehensive characterization of the eigenstructure
Complex eigenvalues and eigenvectors may arise in certain cases
Require computations in the complex field
Represent rotations combined with scalings in the plane spanned by real and imaginary parts of corresponding eigenvectors
Eigenvalues with absolute value greater than 1 indicate expansion along eigenvector direction
Eigenvalues with absolute value less than 1 signify contraction along eigenvector direction
Computing Eigenvalues and Eigenvectors
Characteristic Equation and Eigenvalues
Construct characteristic equation det(T−λI)=0 to find eigenvalues
T represents matrix of linear operator
I denotes identity matrix
Solve characteristic equation to obtain eigenvalues
Characteristic equation forms a polynomial equation in λ
Roots of equation correspond to eigenvalues of linear operator T
Degree of characteristic polynomial equals dimension of vector space on which T acts
Coefficient of highest degree term in characteristic polynomial always equals (−1)n
n represents dimension of space
Constant term of characteristic polynomial equals (−1)n times determinant of T
Eigenvector Computation
Find non-trivial solutions to equation (T−λI)v=0 for each eigenvalue λ to obtain corresponding eigenvectors
Employ Gaussian elimination or other matrix methods to solve homogeneous system (T−λI)v=0 for each eigenvalue