Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Again, by the spectral theorem, such a matrix takes the general form:

This is the spectral theorem in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.

It follows from the spectral theorem for compact self-adjoint operators that every mixed state is an infinite convex combination of pure states.

The Wiener – Khinchin theorem, ( or Wiener – Khintchine theorem or Khinchin – Kolmogorov theorem ), states that the power spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding autocorrelation function.

The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free digital data ( that is, information ) that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density.

In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about 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 ).

In general, the spectral theorem identifies a class of linear operators that can be modelled 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.

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.

In this article we consider mainly 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.

The finite-dimensional spectral theorem says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix.

Another way to phrase the spectral theorem is that a real n × n matrix A is symmetric if and only if there is an orthonormal basis of consisting of eigenvectors for A.

* U is diagonalizable ; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem.

In the special case that M is a normal matrix, which by definition must be square, the spectral theorem says that it can be unitarily diagonalized using a basis of eigenvectors, so that it can be written for a unitary matrix U and a diagonal matrix D. When M is also positive semi-definite, the decomposition is also a singular value decomposition.

An invertible linear transformation applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable rotation, a consequence of the polar decomposition ( also, see spectral theorem ).

If the linear transformation is represented by a symmetric 3-by-3 matrix, then the eigenvectors of the matrix are orthogonal ( due to the spectral theorem ) and represent the directions of the axes of the ellipsoid: the lengths of the semiaxes are given by the eigenvalues.

The spectral theorem can again be used to obtain a standard equation akin to the one given above.

In accordance with the spectral theorem, it is thus possible to diagonalise the tensor by choosing the appropriate set of coordinate axes, zeroing all components of the tensor except χ < sub > xx </ sub >, χ < sub > yy </ sub > and χ < sub > zz </ sub >.

The stellar classification of this star is B2. 5, with numerous spectral lines suggesting it has been polluted by matter ejected by Antares.

The Morgan-Keenan spectral classification

He was studying the relationship between the spectral classification of stars and their actual brightness as corrected for distance — their absolute magnitude.

The MK classification assigned each star a spectral type — based on the Harvard classification — and a luminosity class.

The values of luminosity ( L ), radius ( R ) and mass ( M ) are relative to the Sun — a dwarf star with a spectral classification of G2 V. The actual values for a star may vary by as much as 20 – 30 % from the values listed below.

The general spectral type of Rigel as B8 is well established and it has been used as a defining point of the spectral classification sequence for supergiants.

In astronomy, stellar classification is a classification of stars based on their spectral characteristics.

Under the Morgan-Keenan spectral classification scheme, planetary nebulae are classified as Type-P, although this notation is seldom used in practice.

Draw some conclusions: for example, the SGXB SAX J1818. 6-1703 was discovered by BeppoSAX in 1998, identified as a SGXB of spectral type between O9I − B1I, which also displayed short and bright flares and an unusually very low quiescent level leading to its classification as a SFXT.

The star spectral classification and discovery of the main sequence, Hubble's law and the Hubble sequence were all made with spectrographs that used photographic paper.

Perhaps his greatest contribution to astronomy was the development of a classification system for stars to divide them by spectral type, stage in their development, and luminosity.

( See spectral classification for more information.

It was inspired by the Yerkes spectral classification system for describing stars.

Unlike the Geek Code, this spectral classification system uses classes, subclasses & peculiarities for categorization.

In 1918 this star appeared in the Henry Draper Catalogue with the designation HD 22049 and a preliminary spectral classification of K0.

In the Yerkes spectral classification, supergiants are class Ia ( most luminous supergiants ) or Ib ( less luminous supergiants ).

* MK spectral classification, a stellar classification system based on spectral lines

Wolf 359 has a stellar classification of M6. 5, although various sources list a spectral class of M5. 5, M6 or M8.

The norm of a normal element x of a C *- algebra coincides with its spectral radius.

In normal human vision, the spectral sensitivity of a cone falls into one of three subgroups.

The concept of normality is important because normal matrices are precisely those to which the spectral theorem applies: a matrix A is normal if and only if it can be represented by a diagonal matrix Λ and a unitary matrix U by the formula

Another way of stating the spectral theorem is to say that normal matrices are precisely those matrices that can be represented by a diagonal matrix with respect to a properly chosen orthonormal basis of C < sup > n </ sup >.

The spectral theorem for normal matrices can be seen as a special case of the more general result which holds for all square matrices: Schur decomposition.

The operator norm of a normal matrix N equals the spectral and numerical radii of N. ( This fact generalizes to normal operators.

The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix ( if AA < sup >*</ sup > = A < sup >*</ sup > A then there exists a unitary matrix U such that UAU < sup >*</ sup > is diagonal ).

Every Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies.

This implies the usual spectral theorem: every normal operator on a finite-dimensional space is diagonalizable by a unitary operator.

The operator norm of a normal operator equals its numerical radius and spectral radius.

The spectral theorem still holds for unbounded normal operators, but usually requires a different proof.

* A Fabry – Pérot etalon can be used to make a spectrometer capable of observing the Zeeman effect, where the spectral lines are far too close together to distinguish with a normal spectrometer.

By spectral theorem, a bounded operator on a Hilbert space is normal if and only if it is a multiplication operator.

If X is a Hilbert space and T is a normal operator, then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators ( Hermitian matrices, for example ).

This can be seen as a consequence of the spectral theorem for normal operators.

The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix.

In the latter case, the Gelfand-Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix.