Spectrum of a matrix
Definition
In linear algebra, the spectrum of a matrix is the set of its eigenvalues. The term "spectrum" is derived from the Latin word "spectra", meaning "image" or "appearance". It is a fundamental concept in various branches of mathematics such as matrix theory, functional analysis, and quantum mechanics.
Mathematical Background
The concept of the spectrum of a matrix is closely related to the concept of eigenvalues and eigenvectors. An eigenvalue of a square matrix A is a scalar λ such that there exists a non-zero vector v satisfying the equation Av = λv. The vector v is called an eigenvector corresponding to the eigenvalue λ. The set of all eigenvalues of a matrix A is called the spectrum of A, denoted by σ(A).
Properties
The spectrum of a matrix has several important properties:
1. The spectrum of a matrix is always a subset of the complex numbers. 2. The sum of the eigenvalues (including multiplicities) is equal to the trace of the matrix, which is the sum of the diagonal elements of the matrix. 3. The product of the eigenvalues (including multiplicities) is equal to the determinant of the matrix. 4. The spectrum of a diagonal matrix is just the set of its diagonal elements. 5. The spectrum of a matrix is invariant under similarity transformations. That is, if A and B are similar matrices, then σ(A) = σ(B).
Applications
The spectrum of a matrix plays a crucial role in many areas of mathematics and its applications. For instance, in quantum mechanics, the eigenvalues of a Hamiltonian operator represent the possible energy levels of a quantum system. In graph theory, the spectrum of the adjacency matrix of a graph provides information about the graph's structure. In control theory, the stability of a system can often be determined by examining the spectrum of its system matrix.
See Also
Eigenvalues and Eigenvectors Matrix Theory Functional Analysis Quantum Mechanics Graph Theory Control Theory