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In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. [1]
The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.
In mathematics, the spectrum of a matrix is the set of its eigenvalues. [ 1 ] [ 2 ] [ 3 ] More generally, if T : V → V {\displaystyle T\colon V\to V} is a linear operator on any finite-dimensional vector space , its spectrum is the set of scalars λ {\displaystyle \lambda } such that T − λ I {\displaystyle T-\lambda I} is not invertible .
In social science, economic spectrum is used to indicate the range of social class along some indicator of wealth or income. In political science , the term political spectrum refers to a system of classifying political positions in one or more dimensions, for example in a range including right wing and left wing.
Spectral signature is the variation of reflectance or emittance of a material with respect to wavelengths (i.e., reflectance/emittance as a function of wavelength). [1] The spectral signature of stars indicates the composition of the stellar atmosphere .
The classical example of a discrete spectrum (for which the term was first used) is the characteristic set of discrete spectral lines seen in the emission spectrum and absorption spectrum of isolated atoms of a chemical element, which only absorb and emit light at particular wavelengths. The technique of spectroscopy is based on this phenomenon.
An arbitrary intersection of compact and open subsets of X (hence of elements from K (X)) is again spectral. X is T 0 by definition, but in general not T 1. [1] In fact a spectral space is T 1 if and only if it is Hausdorff (or T 2) if and only if it is a boolean space if and only if K (X) is a boolean algebra.
This vector would contain the brightness values for each pixel in each band in each training class. The mean, standard deviation, variance-covariance matrix, and correlation matrix are calculated from the measurement vectors. Once the statistics from each training site are determined, the most effective bands for each class should be selected.