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...nt +1. Since for such (non-identity) matrices exactly one eigenvector has eigenvalue +1, this eigenvector gives the direction of the axis. ...that the vector '''n''' is an [[eigenvector]] of the matrix '''R''' with eigenvalue λ = 1.

* Corresponding to each eigenvalue λ_''n'' is a unique (up to a normalization constant) eigen ...neralized operator eigenvalue equation. It can be transformed to a regular eigenvalue equation by substitution of

Eigenfunctions of the [[Fourier Operator]] with eigenvalue unity are also eigenfunctions of the [[CosFourier]]. Such functions can be

...r these points the potential energy ''V''(''x'') is larger than the energy eigenvalue. As an example two small vertical lines are plotted: to the left of the sm ...ative, so that ''V'' ≤ ''E''. This is why the region, where the energy eigenvalue of an eigenfunction is smaller than the potential energy, is called a ''cla

...the differential equation has the form of an [[eigenvalue equation]] with eigenvalue −''n''(''n''+1) of the operator

...can be done because the two equations are equivalent to the [[generalized eigenvalue problem]] ...[numerical mathematics]] several algorithms are known to solve generalized eigenvalue problems. The solution yields the matrix '''L''' and the diagonal matrix ''

...m, expressed in displacement coordinates, one must solve the [[generalized eigenvalue problem]] ...diagonal matrix <math>\scriptstyle \boldsymbol{\Phi}</math> contains the [[eigenvalue]]s, while

...wever, the two polynomials have the same set of roots, namely the set of [[eigenvalue]]s of ''A''.

...function; id set, $~S~$ is [[eigenfunction]] of the Fourier operator with eigenvalue unity. ...self-Fourier functions are [[eigenfunction]] of the Fourier operator with eigenvalue unity.

...sharp. This means that the observable takes a value which is one of the [[Eigenvalue|eigenvalues]] of the Hermitian matrix. This set of values is the observable

...sup>2</sup> has only non-negative real eigenvalues. Therefore we write its eigenvalue as the squared number ''a''². ...h>\scriptstyle a^2</math> and an eigenvector of ''j''_''z'' with eigenvalue ''b'' + 1, so that

...this molecule, is a [[partial differential equation|partial differential eigenvalue equation]] in 162 variables&mdash;the spatial coordinates of ''the electron ...'' stands for all electronic coordinates. Obviously, the electronic energy eigenvalue ''E''_elec depends on the chosen positions '''R''' of the nuclei.

...s of the time-independent [[Schrödinger equation]] of molecules&mdash;the eigenvalue equation of a [[molecular Hamiltonian]] (quantum mechanical energy operato ...h is not surprising, as the corresponding HF operator equation is also an eigenvalue equation. The matrix to be diagonalized (known as the ''Fock matrix'' '''F

...''n'' roots (zeros) &lambda;_''i'', ''i'' =1, ..., ''n'', the [[eigenvalue]]s of '''A''' and the following factorization holds ...s &epsilon;_''i''. When ''H'' is bounded from below (its lowest eigenvalue is finite) then the sum

...hat the fact that the time independent equation takes the form of a simple eigenvalue problem (thus being more amenable to mathematical analysis) that makes it s ...matically, the time-independent Schrödinger equation is an example of an [[eigenvalue problem]] whose eigenvectors &psi; are called "wavefunctions" or "quantum

1. The [[eigenvalue|eigenvalues]] of a self-adjoint operator are real. As a special well-known

...nics]], '''angular momentum coupling''' is the procedure of constructing [[eigenvalue|eigenvectors]] of a system's [[angular momentum (quantum)|angular momentum ...em can assume a sharp (i.e., eigen-) value simultaneously with the energy (eigenvalue of the Hamiltonian) of the system.

...ponents. It is conventional to choose ''l''_''z'', which has an eigenvalue proportional to an integer usually denoted by ''m'' (the so-called [[magne The Schrödinger equation is the following eigenvalue equation of the Hamiltonian (the quantity in large square brackets):

...ng but the [[eigenvalue problem]], and our task is to determine the energy eigenvalue(s) <math>E</math> and [[eigenstate]](s) <math>\psi</math> that solve it. To

...'spherical well problem''' is to find the [[energy]] [[eigenstates]] and [[eigenvalue|values]] of a single [[particle]] of [[mass]] <math>m</math> constrained wi

...ions have determinant &minus;1. Because &sigma; is symmetric it has real [[eigenvalue]]s; since the extension of an isometry to a complex space is unitary, its (

The eigenvalue equation becomes The eigenvalue ''E''_''j''0 is (2''j'' + 1)-fold degenerate, because all eigenf

...lved for a sufficient number of constellations of the nuclei, the set of [[eigenvalue]]s (usually the lowest) can be seen as a [[function]] of the nuclear coor ...ignore spin and other relativistic terms and considers the solution of the eigenvalue (time-independent Schrödinger) equation of the Coulomb Hamiltonian.

...f the [[orbital angular momentum operator]] with quantum number ''l'' = 1 [eigenvalue 1(1+1)]. ...of order ''l'', which are eigenfunctions of orbital angular momentum with eigenvalue ''l''(''l''+1). The quantum number ''l'' runs as follows

The eigenvalue equation can be simplified by [[separation of variables]]. We substitute into the eigenvalue equation. After dividing out &Psi; and multiplying with sin&sup2;&theta; we

...he self-Fourier function is eigenfunction of the [[Fourier operator]] with eigenvalue 1;

...hat a rotation is characterized by a unique axis (an eigenvector with unit eigenvalue) and a unique angle. As a corollary of the theorem follows that the [[trace The eigenvalue equation can be transformed

...mentum is ℏ'''k''' [ℏ=''h''/(2π)]; the polarization &mu;=&plusmn;1 is the eigenvalue of the ''z''-component of the photon spin. ...hoton state is an eigenstate of the momentum operator, and ℏ'''k''' is the eigenvalue (the momentum of a single photon).

...bosons must be an eigenfunction of a transposition operator with unity as eigenvalue

...]]-[[Fock]]-[[Roothaan]] equations, which have the form of a generalized [[eigenvalue problem]]. The dimension of this problem is equal to the number of atomic o

...ear operator]]. In this case, the fixed point is an [[eigenvector]] with [[eigenvalue]] equal to unity. For instance, the [[exponential function]] is a fixed poi

...erminant is an eigenfunction of all three angular momentum components with eigenvalue zero, i.e., the total angular momentum quantum number of the Slater determi

...s must be an eigenfunction of a transposition operator with its parity as eigenvalue

...ef> E. Schrödinger, ''Quantisierung als Eigenwerproblem'' [quantization as eigenvalue problem] Annalen der Physik, Vierte Folge, Band 79, p. 361 (1926)

...eigenvectors (the group theoretical functions) have been found. The first eigenvalue (energy of the state of ''T''_2''g'' symmetry) is 3-fold degenera

.../sub> appears linearly in the lowest first-order energy correction (lowest eigenvalue of the perturbation matrix); terms of higher order in the infinitesimal coo

...'n''</sub> of a quantum mechanical system appear as [[eigenvalues]] of the eigenvalue equation In quantum mechanical studies the eigenvalue problem of ''any'' observable may appear occasionally. However, the observa

*[[User:Michael Underwood|Michael]] wrote about [[eigenvalue]]s and tidied up the laws of [[law of cosines|cosines]] and [[law of sines|

*''Comparison of Eigenvalue, Logarithmic Least Squares and Least Squares Methods in Estimating Ratios,'

...le corresponds to an [[eigenvector]] of the operator, and the associated [[eigenvalue]] corresponds to the value of the observable in that eigenstate. If the ope

and <math>L</math> is eigenvalue of logarithm, solution of equation