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Jitse Niesen (talk | contribs) →Spectral theorem: revert - P is applied to T, not to Tv. Also, be pedantic: the formula is not a P(T), but P(T)v |
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where the juxtaposition is [[matrix multiplication]]. Since, once a basis is fixed, ''T'' and its matrix representation ''A''<sub>''T''</sub> are equivalent, we can often use the same symbol ''T'' for both the matrix representation and the transformation. This is equivalent to a set of ''n'' linear equations, where ''n'' is the number of basis vectors in the [[basis (linear algebra)|basis set]]. In this equation both the eigenvalue ''λ'' and the ''n'' components of '''v'''<sub>''λ''</sub> are [[variable|unknown]]s.
However, it is sometimes unnatural or even impossible to write down the eigenvalue equation in a matrix form. This occurs for instance when the vector space is infinite dimensional, for example, in the case of the rope above. Depending on the nature of the transformation ''T'' and the space to which it applies, it can be advantageous to represent the eigenvalue equation as a set of [[differential equation]]s. If ''T'' is a [[differential operator]], the eigenvectors are commonly called '''eigenfunctions''' of the differential operator representing ''T''. For example, [[Differential calculus|differentiation]] itself is a linear transformation since
:<math> \displaystyle\frac{d}{dt}(af+bg) = a \frac{df}{dt} + b \frac{dg}{dt} </math>
(''f''(''t'') and ''g''(''t'') are [[differentiable]] functions, and ''a'' and ''b'' are [[constant]]s).
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==Infinite-dimensional spaces==
[[Image:Discrete-continuum.png|thumb|250px|Fig. 3.[[absorption spectroscopy|Absorption]] [[spectroscopy|spectrum]] ([[cross section (physics)|cross section]]) of atomic [[Chlorine]]. The sharp lines obtained in theory correspond to the [[discrete spectrum]] ([[Rydberg atom|Rydberg series]]) of the [[Hamiltonian (quantum mechanics)|Hamiltonian]]; the broad structure on the right is associated with the [[continuous spectrum]] ([[ionization]]). The corresponding [[experiment]]al results have been obtained by measuring the intensity of [[X-ray]]s absorbed by a gas of atoms as a function of the incident [[photon]] [[energy]] in [[Electronvolt|eV]].<ref>T. W Gorczyca, Auger Decay of the Photoexcited Inner Shell Rydberg Series in Neon, Chlorine, and Argon, Abstracts of the 18th International Conference on X-ray and Inner-Shell Processes, Chicago, August 23-27 (1999).</ref>]]
If the vector space is an infinite dimensional [[Banach space]], the notion of eigenvalues can be generalized to the concept of [[spectrum]]. The spectrum is the set of scalars λ for which <math>\left(T-\lambda\right)^{-1}</math> is not defined; that is, such that <math>T-\lambda</math> has no [[bounded operator|bounded]] inverse.
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where ''H'', the [[Hamiltonian (quantum mechanics)|Hamiltonian]], is a second-order [[differential operator]] and <math>\psi_E</math>, the [[wavefunction]], is one of its eigenfunctions corresponding to the eigenvalue ''E'', interpreted as its [[energy]].
However, in the case where one is interested only in the [[bound state]] solutions of the Schrödinger equation, one looks for <math>\psi_E</math> within the space of [[square integrable]] functions. Since this space is a [[Hilbert space]] with a well-defined [[scalar product]], one can introduce a [[Basis (linear algebra)|basis set]] in which <math>\psi_E</math> and ''H'' can be represented as a one-dimensional array and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form. (Fig. 4 presents the lowest eigenfunctions of the [[Hydrogen atom]] Hamiltonian.)
The [[Dirac notation]] is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by <math>|\Psi_E\rangle</math>. In this notation, the Schrödinger equation is:
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===Stress tensor===
In [[solid mechanics]], the [[stress tensor]] is symmetric and so can be decomposed into a [[diagonal]] tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no [[Shear (mathematics)|shear]] components; the components it does have are the principal components.
===Eigenvalues of a graph===
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