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 >.

A set of tables containing spectral intensities for 39,000 lines of 70 elements, as observed in a copper matrix in a d-c arc, was completed and published.

It was possible to make estimates of the quantum yield by observing the extent of reduction of a uranyl oxalate actinometer solution illuminated for a known time in a typical reaction cell and making appropriate conversions based on the differences in the absorption spectra of uranyl oxalate and of chlorine, and considering the spectral distribution of the light source.

The same consideration should govern the choice of the second-stage phosphor screen for matching with the spectral sensitivity of the ultimate sensor ( e.g., photographic emulsion ).

Atle Selberg ( 14 June 1917 – 6 August 2007 ) was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory.

The alpha component may be used to blend to red, green and blue components equally, as in 32-bit RGBA, or, alternatively, there may be three alpha values specified corresponding to each of the primary colors for spectral color filtering.

More alpha channels can be added for accurate spectral color filtration applications.

In classical LS AAS, as it has been proposed by Alan Walsh, the high spectral resolution required for AAS measurements is provided by the radiation source itself that emits the spectrum of the analyte in the form of lines that are narrower than the absorption lines.

In CS AAS, in contrast, a single lamp, emitting a continuum spectrum over the entire spectral range of interest is used for all elements.

The same algorithm can actually also be used to correct for direct line overlap of two atomic absorption lines, making HR-CS AAS the only AAS technique that can correct for this kind of spectral interference.

Thomson attempted to make his model account for some of the major spectral lines known for some elements, but was not notably successful at this.

Its spectral type is A0p ; the " p " stands for peculiar, as the spectrum of its light is characteristic of an Alpha2 Canum Venaticorum variable.

In radio communications, for example, bandwidth is the frequency range occupied by a modulated carrier wave, whereas in optics it is the width of an individual spectral line or the entire spectral range.

This same " half power gain " convention is also used in spectral width, and more generally for extent of functions as full width at half maximum ( FWHM ).

Larry Gragg highlights Mather ’ s cloudy thinking and confusion between sympathy for the possessed, and the boundlessness of spectral evidence when Mather stated, “ the devil have sometimes represented the shapes of persons not only innocent, but also the very virtuous .” And writing in the early 1980s, John Demos seemed to consider Mather a moderating influence on the trials.

This relation, however, is a psychological one in contrast to the physical relation implied by Wien's displacement law, according to which the spectral peak is shifted towards shorter wavelengths ( resulting in a more blueish white ) for higher temperatures.

The K-corona ( K for kontinuierlich, " continuous " in German ) is created by sunlight scattering off free electrons ; Doppler broadening of the reflected photospheric absorption lines completely obscures them, giving the spectral appearance of a continuum with no absorption lines.

The E-corona ( E for emission ) is due to spectral emission lines produced by ions that are present in the coronal plasma ; it may be observed in broad or forbidden or hot spectral emission lines and is the main source of information about the corona's composition.

DSP includes subfields like: audio and speech signal processing, sonar and radar signal processing, sensor array processing, spectral estimation, statistical signal processing, digital image processing, signal processing for communications, control of systems, biomedical signal processing, seismic data processing, etc.

After leaving the experiment running for a long period of time a spectral analysis of the contents of the former evacuated space revealed the presence of helium.

Both the UVS and EUV instruments used a ruled grating to disperse this light for spectral analysis.