Properties of Eigenvalues

The sum of the eigenvalues of a matrix equals the trace of the matrix the trace of a matrix being the sum of the elements on the principal, or main, diagonal. 

The product of the eigenvalues of a matrix equals the determinant of the matrix, thus, 

Carrying out the multiplications required by Eq. A.4.15 and solving for 2 in terms of e gives 

Since the eigenvectors are solutions of a homogeneous system of equations, the solution is determined only up to a constant factor and only the ratios of the elements in the columns (e ) are uniquely determined. The geometrical interpretation of this is that the eigenvectors are uniquely determined only in their direction, but their length or absolute value is arbitrary. 

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Let us dwell on the properties of eigenvalues and eigenvectors of a linear self-adjoint operator A. A number A such that there exists a vector 0 with = A is called an eigenvalue of the operator A. This vector... 

PROPERTIES OF EIGENVALUES AND EIGENVECTORS Equation (1.44) is easily verified ... 

The following are some general properties of eigenvalues worth not-... 

A general property of eigenvalue equations such as the time-independent Schrodinger equation is that if is a solution, then so is the product of with any constant. This means that we can multiply any ntm-zero eigenfunction by a... 

Kaa is the diagonal matrix derived from the classical stiffness matrix by condensation on the reduced modal space. For example, if the chosen modes are the body eigenmodes in vibration, according to the property of eigenvalues, this matrix reduces to the diagonal matrix with the squares of the naturi frequencies of the modes of vibration selected. 

To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... 

The last identity follows from the orthogonality property of eigenfunctions and the assumption of nomralization. The right-hand side in the final result is simply equal to the sum over all eigenvalues of the operator (possible results of the measurement) multiplied by the respective probabilities. Hence, an important corollary to the fiftli postulate is established ... 

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). 

In quantum mechanics, the eigenvalues of an operator represent the only numerical values that can be observed if the physical property corresponding to that operator is measured. Operators for which the eigenvalue spectrum (i.e., the list of eigenvalues) is discrete thus possess discrete spectra when probed experimentally. 

According to the Floquet theorem [Arnold 1978], this equation has a pair of linearly-independent solutions of the form x(z,t) = u(z, t)e p( 2nizt/p), where the function u is -periodic. The solution becomes periodic at integer z = +n, so that the eigenvalues e we need are = ( + n). To find the infinite product of the we employ the analytical properties of the function e z). It has two simple zeros in the complex plane such that... 

Eigenvalues of J and J2.—From the defining relations (7-18), certain very general properties of all angular momenta can be derived in particular, it is possible to obtain a catalogue and a classification of all possible eigenvalues of Jz and of J2. 

Applying the hermitian property of and noting that is orthogonal to all eigenfunctions belonging to the eigenvalue we have... 

This dependence of the TS trajectory on the past and future driving is illustrated in Fig. 2, which shows the time dependence of the 5-functional 5[p, (t) for a fixed driving field t) and different values of the eigenvalue p. To demonstrate the properties of the 5-functional clearly, we have chosen a smooth driving field (t) that is given by Eq. (81) (with Ao = 1, co = 1, and N = 2). It is zero for f > 27i. 

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