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Normal Forms and Inverse Spectral Theory
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People also ask
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What is the spectral theory of functional analysis?
In 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).
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What is the spectral theory of a normed linear space?
Spectral theory is the study of the distribution of the values of the complex parameter λ for which, given a linear operator A on a normed space E, the operator A − λI has an inverse and of the properties of this inverse when it exists, the resolvent R(A,λ) = (A − λI)−1 of A.
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What is the spectral theory of normal operators?
In summary, a matrix is normal if and only if it can be written as a linear combination of orthogonal idempotents Eλ such that EλEµ = 0 when λ 6= µ and Pλ Eλ = 1. This is called the spectral theorem for normal operators and the above expression for the operator is called the spectral decomposition of the operator.
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What is the spectral theory of linear operators?
The principal task of the spectral theory of linear operators is to determine the set of allowable complex numbers λ, which is called the resolvent set, assuring the existence of a bounded inverse of the operator A λ and to illuminate the relations between this set and its complement in the complex plane called the ...
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What is the spectral theorem explained?
The spectral theorem provides a sufficient criterion for the existence of a particular canonical form. Specifically, the spectral theorem states that if M equals the transpose of M, then M is diagonalizable: there exists an invertible matrix C such that C − 1 M C C^{-1} MC C−1MC is a diagonal matrix.
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What is the spectral theory of normed linear spaces?
Spectral theory is the study of the distribution of the values of the complex parameter λ for which, given a linear operator A on a normed space E, the operator A − λI has an inverse and of the properties of this inverse when it exists, the resolvent R(A,λ) = (A − λI)−1 of A.
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What is the spectral theory of number theory?
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.
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What is the spectral theory of odes?
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation.
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