This is crucial in assessing the sensitivity and potential accuracy difficulties of numerous computational problems, for example polynomial root finding or computing eigenvalues.

Since there are only four distinct eigenvalues for this matrix, they have some multiplicity.

As terms of the form λ < sub > i </ sub > – ∑ ( multiples of other eigenvalues ) occurs in the denominator of the terms for the function h, the non-resonant condition is also known as the small divisor problem.

The possible results of a measurement are the eigenvalues of the operator representing the observable — which explains the choice of Hermitian operators, for which all the eigenvalues are real.

But in fact any system possessing an observable quantity A which is conserved under time evolution and such that A has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit.

Attempted replacements for the Schrödinger equation, such as the Klein-Gordon equation or the Dirac equation, have many unsatisfactory qualities ; for instance, they possess energy eigenvalues that extend to –∞, so that there seems to be no easy definition of a ground state.

Thus, for a substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers.

It has the property that the eigenvalues and eigenvectors determine the physical quantity and the states which have definite values for this quantity.

Eigenvectors corresponding to other eigenvalues are orthogonal to, so for such eigenvectors, we have.

If, for whatever reason, two eigenvalues are close so that difference become small, the corresponding term in the sum will become disproportionately large.

Having determinant ± 1 and all eigenvalues of magnitude 1 is of great benefit for numeric stability.

Solve for r to obtain the two roots λ < sub > 1 </ sub >, λ < sub > 2 </ sub >: these roots are known as the characteristic roots or eigenvalues of the characteristic equation.

In the case of complex eigenvalues ( which also gives rise to complex values for the solution parameters C and D ), the use of complex numbers can be eliminated by rewriting the solution in trigonometric form.

The stability condition stated above in terms of eigenvalues for the second-order case remains valid for the general n < sup > th </ sup >- order case: the equation is stable if and only if all eigenvalues of the characteristic equation are less than one in absolute value.

* A number whose discrete values ( eigenvalues ) are the positive roots of transcendental equations, used in the series solutions for transient one-dimensional conduction equations.

# for all where A has singular values and eigenvalues

The formula for area eigenvalues becomes somewhat more complicated if the surface is allowed to pass through the nodes ( it is not yet clear if these situations are physically meaningful.

This holds for all possible eigenvalues λ, so the two matrices equated by the theorem certainly give the same ( null ) result when applied to any eigenvector.

Solving for the eigenvalues of this matrix, ( as can be done by hand, or more easily, with a computer algebra system ) we arrive at the energy shifts:

In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues ; more precisely: as spectral values ( point spectrum plus absolute continuous plus singular continuous spectrum ) of linear operators in Hilbert space.

Applying these ladder operators to the eigenstates of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator C < sub > 1 </ sub > are n < sup > 2 </ sup > − 1 ; importantly, they are independent of the l and m quantum numbers, making the energy levels degenerate.

The angular momentum states must be orthogonal ( because their eigenvalues with

Since the partial derivative is a linear operator, the momentum operator is also linear, and because any wavefunction can be expressed as a superposition principle of other states, when this momentum operator acts on the entire superimposed wave, it yields the momentum eigenvalues for each plane wave component, the momenta add to the total momentum of the superimposed wave.

For example, in the hydrogen atom, for a fixed energy eigenvalue, there exist several states which have that energy, but differ in the eigenvalues of angular momentum, spin component and so on.

Clearly, P² = 1 ( the identity operator ), so the eigenvalues of P are + 1 and − 1.

The " ladder operator " method, due to Paul Dirac, allows us to extract the energy eigenvalues without directly solving the differential equation.

Another postulate of quantum mechanics is that all observables are represented by operators which act on the wavefunction, and the eigenvalues of the operator are the values the observable takes.

The principal curvatures are the eigenvalues of the shape operator, and in fact the shape operator and second fundamental form have the same matrix representation with respect to a pair of orthonormal vectors of the tangent plane.

The von Neumann entropy of a mixture can be expressed in terms of the eigenvalues of or in terms of the trace and logarithm of the density operator.

The degeneracy of eigenvalues indicates that the unperturbed system has some sort of symmetry, and that the generators of the symmetry commute with the unperturbed differential operator.

The spectrum of any bounded symmetric operator is real ; in particular all its eigenvalues are real, although a symmetric operator may have no eigenvalues.

If the set of eigenvalues for a symmetric operator is non empty, and the eigenvalues are nondegenerate, then it follows from the definition that eigenvectors corresponding to distinct eigenvalues are orthogonal.

Contrary to what is sometimes claimed in introductory physics textbooks, it is possible for symmetric operators to have no eigenvalues at all ( although the spectrum of any self-adjoint operator is nonempty ).

A is a symmetric operator without any eigenvalues and eigenfunctions.

is an eigenvalue of a normal operator N if and only if its complex conjugate is an eigenvalue of Eigenvectors of a normal operator corresponding to different eigenvalues are orthogonal, and it stabilizes orthogonal complements to its eigenspaces

In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices.

The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues.

However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues.

In other words, the spectrum of such an operator, which was defined as a generalization of the concept of eigenvalues, consists in this case only of the usual eigenvalues, and possibly 0.

Spurred by the need for enough financial security to marry, Bush finished his thesis, entitled Oscillating-Current Circuits: An Extension of the Theory of Generalized Angular Velocities, with Applications to the Coupled Circuit and the Artificial Transmission Line, in April 1916.

Constant Angular Acceleration is very similar to Constant Linear Velocity, save for the fact that CAA varies the angular rotation of the disc in controlled steps instead of gradually slowing down in a steady linear pace as a CLV disc is read.

* Constant Angular Velocity, a qualifier for the rated speed of an optical disc drive

Angular dependent calculations or measurements on systems consisting of a single crystal of a compound, for example, are anisotropic, meaning the density of states will be different in one crystallographic direction than in another.

Angular inertia determines the rotational inertia of an object for a given rate of rotation.

Angular measurements are averaged over a span of time and combined with radial movement to develop information suitable to predict target position for a short time into the future.

Construction began in 1996 for six 1-meter telescopes by the Center for High Angular Resolution Astronomy at Georgia State University.

The CHARA Array is an optical astronomical interferometer operated by The Center for High Angular Resolution Astronomy ( CHARA ) of the Georgia State University ( GSU ).