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In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or matrices. In broad terms, the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). Intuitively, diagonal matrices are computationally quite manageable, so it is of interest to see whether an arbitrary matrix can be diagonalized. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C *-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts. Augustin Louis Cauchy proved the spectral theorem for self-adjoint matrices, i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systematic about determinants.〔(Cauchy and the spectral theory of matrices by Thomas Hawkins )〕〔(A Short History of Operator Theory by Evans M. Harrell II )〕 The spectral theorem as generalized by John von Neumann is today perhaps the most important result of operator theory. This article mainly focuses on the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space. == Finite-dimensional case == === Hermitian maps and Hermitian matrices === We begin by considering a Hermitian matrix on or . More generally we consider a Hermitian map on a finite-dimensional real or complex inner product space endowed with a positive definite Hermitian inner product. The Hermitian condition means that for all , : An equivalent condition is that where is the hermitian conjugate of . In the case that is identified with an Hermitian matrix, the matrix of can be identified with its conjugate transpose. If is a real matrix, this is equivalent to (that is, is a symmetric matrix). This condition easily implies that all eigenvalues of a Hermitian map are real: it is enough to apply it to the case when is an eigenvector. (Recall that an eigenvector of a linear map is a (non-zero) vector such that for some scalar . The value is the corresponding eigenvalue. Moreover, the eigenvalues are solutions to the characteristic polynomial.) Theorem. There exists an orthonormal basis of consisting of eigenvectors of . Each eigenvalue is real. We provide a sketch of a proof for the case where the underlying field of scalars is the complex numbers. By the fundamental theorem of algebra, applied to the characteristic polynomial of , there is at least one eigenvalue and eigenvector . Then since : we find that is real. Now consider the space , the orthogonal complement of . By Hermiticity, is an invariant subspace of . Applying the same argument to shows that has an eigenvector . Finite induction then finishes the proof. The spectral theorem holds also for symmetric maps on finite-dimensional real inner product spaces, but the existence of an eigenvector does not follow immediately from the fundamental theorem of algebra. The easiest way to prove it is probably to consider as a Hermitian matrix and use the fact that all eigenvalues of a Hermitian matrix are real. If one chooses the eigenvectors of as an orthonormal basis, the matrix representation of in this basis is diagonal. Equivalently, can be written as a linear combination of pairwise orthogonal projections, called its spectral decomposition. Let : be the eigenspace corresponding to an eigenvalue . Note that the definition does not depend on any choice of specific eigenvectors. is the orthogonal direct sum of the spaces where the index ranges over eigenvalues. Let be the orthogonal projection onto and the eigenvalues of , one can write its spectral decomposition thus: : The spectral decomposition is a special case of both the Schur decomposition and the singular value decomposition. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「spectral theorem」の詳細全文を読む スポンサード リンク
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