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In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix.
In mathematics, the spectrum of a matrix is the set of its eigenvalues. [ 1 ] [ 2 ] [ 3 ] More generally, if T : V → V {\displaystyle T\colon V\to V} is a linear operator on any finite-dimensional vector space , its spectrum is the set of scalars λ {\displaystyle \lambda } such that T − λ I {\displaystyle T-\lambda I} is not invertible .
While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one. Spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the Colin de Verdière number .
The suspension spectrum of a space , denoted is a spectrum = (the structure maps are the identity.) For example, the suspension spectrum of the 0-sphere is the sphere spectrum discussed above. The homotopy groups of this spectrum are then the stable homotopy groups of X {\displaystyle X} , so
The spectrum of T restricted to H ac is called the absolutely continuous spectrum of T, σ ac (T). The spectrum of T restricted to H sc is called its singular spectrum, σ sc (T). The set of eigenvalues of T is called the pure point spectrum of T, σ pp (T). The closure of the eigenvalues is the spectrum of T restricted to H pp.
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