Characteristic polynomial
Definition
The Characteristic Polynomial is a mathematical function associated with a square matrix in linear algebra. It is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has profound implications in the study of linear algebra and plays a pivotal role in the spectral theory.
Mathematical Description
The characteristic polynomial of an n×n matrix A over the field F is defined as the polynomial:
p(λ) = det(A - λI)
where I is the identity matrix of the same size as A, det denotes the determinant, and λ is a scalar in the field F. The polynomial p(λ) is an nth degree monic polynomial in λ, and its roots are the eigenvalues of A.
Properties
The characteristic polynomial has several important properties. These properties are fundamental to many areas of linear algebra and have wide-ranging applications in fields such as differential equations, physics, and engineering.
Invariance under Similarity
If two matrices A and B are similar, then they have the same characteristic polynomial. This is a crucial property as it allows us to study the eigenvalues of a matrix by studying the eigenvalues of a similar matrix, which might be easier to handle.
Roots are Eigenvalues
The roots of the characteristic polynomial are precisely the eigenvalues of the matrix. This property provides a direct method to compute the eigenvalues of a matrix.
Coefficients
The coefficients of the characteristic polynomial have special meanings. The constant term is (−1)n times the determinant of the matrix, and the coefficient of λn−1 is the negative of the trace of the matrix.
Computation
The computation of the characteristic polynomial can be done using various methods. The most straightforward method is to directly compute the determinant of the matrix A - λI. However, this can be computationally expensive for large matrices. Other methods such as the Leverrier-Faddeev algorithm can be used to compute the characteristic polynomial more efficiently.
Applications
The characteristic polynomial has a wide range of applications in various fields of mathematics and science. Some of the key applications are listed below.
Differential Equations
In the field of differential equations, the characteristic polynomial is used to solve linear differential equations with constant coefficients. The roots of the characteristic polynomial give the solutions to the differential equation.
Physics
In physics, the characteristic polynomial is used in the study of quantum mechanics. The eigenvalues of the Hamiltonian matrix, which are the roots of the characteristic polynomial, give the energy levels of a quantum system.
Control Theory
In control theory, the characteristic polynomial is used in the stability analysis of linear systems. The roots of the characteristic polynomial determine the stability of the system.
See Also
Eigenvalues and Eigenvectors Linear Algebra Matrix Theory Quantum Mechanics Control Theory
