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, of the underlying vector space on which the operator acts.
Augustin-Louis Cauchy proved the spectral theorem for symmetric 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 (but the following discussion will be adaptable to the more restrictive case of symmetric matrices on ). We consider a Hermitian map A on a finite-dimensional complex inner product space V endowed with a positive definite sesquilinear inner product .
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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 linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
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