Spectral theorem

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. 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. In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. 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. 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. We begin by considering a Hermitian matrix on C n {displaystyle mathbb {C} ^{n}} (but the following discussion will be adaptable to the more restrictive case of symmetric matrices on R n {displaystyle mathbb {R} ^{n}} ). We consider a Hermitian map A on a finite-dimensional complex inner product space V endowed with a positive definite sesquilinear inner product ⟨ ⋅ , ⋅ ⟩ {displaystyle langle cdot ,cdot angle } . The Hermitian condition on A {displaystyle A} means that for all x, y ∈ V, (An equivalent condition is that A∗ = A, where A∗ is the hermitian conjugate of A.) In the case that A is identified with a Hermitian matrix, the matrix of A∗ can be identified with its conjugate transpose. (If A is a real matrix, this is equivalent to AT = A, that is, A is a symmetric matrix.) This condition implies that all eigenvalues of a Hermitian map are real: it is enough to apply it to the case when x = y is an eigenvector. (Recall that an eigenvector of a linear map A is a (non-zero) vector x such that Ax = λx for some scalar λ. The value λ is the corresponding eigenvalue. Moreover, the eigenvalues are roots of the characteristic polynomial.) Theorem. If A is Hermitian, there exists an orthonormal basis of V consisting of eigenvectors of A. Each eigenvalue is real.