Linear independence

The magnitudes of e i =1, )contam the Fresnel factors from equation Bl.5,34. equation B1,5,35 and equation B 1,5.36. which depend on the incident, reflected and polarization angles. Experimentally, one approach is to fix the input polarization and adjust the analyser to obtain a null in the SFl signal [ ]. By choosing distinct configurations such that the corresponding tliree equations from equation B 1.5.40 are linearly independent, the relative values of Xs lim = inferred. This method has... 

To obtain nonlinear coupling terms, we consider two linearly independent, not identical E modes, namely. 

The diabatic LHSFs are not allowed to diverge anywhere on the half-sphere of fixed radius p. This boundary condition furnishes the quantum numhers n - and each of which is 2D since the reference Hamiltonian hj has two angular degrees of freedom. The superscripts n(, Q in Eq. (95), with n refering to the union of and indicate that the number of linearly independent solutions of Eqs. (94) is equal to the number of diabatic LHSFs used in the expansions of Eq. (95). 

The electronic wave functions of the different spin-paired systems are not necessarily linearly independent. Writing out the VB wave function shows that one of them may be expressed as a linear combination of the other two. Nevertheless, each of them is obviously a separate chemical entity, that can he clearly distinguished from the other two. [This is readily checked by considering a hypothetical system containing four isotopic H atoms (H, D, T, and U). The anchors will be HD - - TU, HT - - DU, and HU -I- DT],... 

Note that only the polynomial factors have been given, since the exponential parts are identical for all wave functions. Of course, any linear combination of the wave functions in Eqs. (D.5)-(D.7) will still be an eigenfunction of the vibrational Hamiltonian, and hence a possible state. There are three such linearly independent combinations which assume special importance, namely,... 

Verify this solution by calculating and substituting A and 5A to prove the equality. We can see that A exists because neither row nor column can be obtained from the other by simple multiplication. They are linearly independent. 

The degree of the least polynomial of a square matr ix A, and henee its rank, is the number of linearly independent rows in A. A linearly independent row of A is a row that eannot be obtained from any other row in A by multiplieation by a number. If matrix A has, as its elements, the eoeffieients of a set of simultaneous nonhomo-geneous equations, the rank k is the number of independent equations. If A = , there are the same number of independent equations as unknowns A has an inverse and a unique solution set exists. If k < n, the number of independent equations is less than the number of unknowns A does not have an inverse and no unique solution set exists. The matrix A is square, henee k > n is not possible. 

The problem of n linear independent nonhomogeneous equations in n real unknowns... 

Linear independence implies that no equation in the set can be obtained by multiplying any other equation in the set by a constant. The n x n matrix populated by n elements Gy... 

What we fomierly called the nonhomogeneous vector (Chapter 2) is zero in the pair of simultaneous nomial equations Eq. set (6-38). When this vector vanishes, the pair is homogeneous. Let us try to construct a simple set of linearly independent homogeneous simultaneous equations. 

Any linearly independent set of simultaneous homogeneous equations we can construct has only the zero vector as its solution set. This is not acceptable, for it means that the wave function vanishes, which is contrai y to hypothesis (the electron has to be somewhere). We are driven to the conclusion that the normal equations (6-38) must be linearly dependent. 

The example demonstrates that not all the B-numbers of equation 5 are linearly independent. A set of linearly independent B-numbers is said to be complete if every B-number of Dis a product of powers of the B-numbers of the set. To determine the number of elements in a complete set of B-numbers, it is only necessary to determine the number of linearly independent solutions of equation 13. The solution to the latter is well known and can be found in any text on matrix algebra (see, for example, (39) and (40)). Thus the following theorems can be stated. 

A sufficient condition for a unique Newton direction is that the matrix of constraint derivatives is of full rank (linear independence... 

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... 

The matrices F and M can be found from straightforward integration of (5.9) with the initial conditions being N linearly independent vectors. Then the quasienergy partition function equals... 

Let Ci(z) be an arbitrary solution to (B.9) which does not necessarily satisfy (B.9). Then it can be represented as a linear combination of exponentially increasing and decreasing linearly independent solutions (B.IO). When cc, only the increasing solution survives after a long time, and one may write... 

Equation (8.90) is non-singular since it has a non-zero determinant. Also the two row and column vectors can be seen to be linearly independent, so it is of rank 2 and therefore the system is controllable. 

Rank of a matrix The rank of a matrix is equal to the number of linearly independent rows or eolumns. The rank ean be found by determining the largest square... 

In the case of d-type orbitals, there are six Cartesian GTOs with pre-exponential factors of x, xy, y, xz, yz and z - Only five are linearly independent, e combi nation... 

Many modem computer codes (e.g. GAUSSIAN98) employ so-called redundant internal coordinates] this means that we use all possible internal coordinates, of which there will generally be more than 3N — 6. Only a maximum of 3M — 6 will be linearly independent, and we essentially throw away the remainder at the end of the full calculation. Here is ethene, done using redundant internal coordinates. 

Due to the linear Independence of occupation indices Tji the concentration-dependent parameters can be determined from the concentration-independent ones. In the special case of interactions limited to the fourth order and nearest neighbors one finds for fcc-based alloys... 

The 1 and 2 block probabilities can be parameterized by any pair of linearly independent 1 and/or 2 block probabilities. For example, choosing the density pi... 

We have shown that the number of linearly independent invariants of degree k under the permutation group H is equal to the coefficient of in the Maclaurin expansion of (1.27). This represents an important special case of a proposition by Th. Molien. ... 

Projections, linearly independent, 293 Propagation, of polymerization, 158 Propane, hydrate, 10, 33, 43, 46, 47 hydrate thermodynamic data and lattice constants, 8 + iodoform system, 99 Langmuir constant, 47 water-hydrogen sulfide ternary system, 53... 

Just as a known root of an algebraic equation can be divided out, and the equation reduced to one of lower order, so a known root and the vector belonging to it can be used to reduce the matrix to one of lower order whose roots are the yet unknown roots. In principle this can be continued until the matrix reduces to a scalar, which is the last remaining root. The process is known as deflation. Quite generally, in fact, let P be a matrix of, say, p linearly independent columns such that each column of AP is a linear combination of columns of P itself. In particular, this will be true if the columns of P are characteristic vectors. Then... 

In ra-space any n + 1 of these are linearly dependent. But unless the matrix is rather special in form (derogatory), there exist vectors vx for which any n consecutive vectors are linearly independent (in possible contrast to the behavior in the limit). In fact, this is true of almost every vector vx. Hence, if... 

We conclude this section by deriving an important property of jointly gaussian random variables namely, the fact that a necessary and sufficient condition for a group of jointly gaussian random variables 9i>- >< to be statistically independent is that E[cpjCpk] = j k. Stated in other words, linearly independent (uncorrelated),46 gaussian random variables are statistically independent. This statement is not necessarily true for non-gaussian random variables. 

The fact that linear independence is a necessary condition for statistical independence is obvious. The sufficiency of the condition can be established by noting that the covariance matrix... 

Theorem A.—The y s and their products yield sixteen linearly independent matrices. 

All other products of y-matrices can, by using the commutation rules, be reduced to one of these sixteen elements. The proof of their linear independence is based upon the fact that the trace of any of these matrices except for the unit matrix, I, is zero. If Tr is any one of these matrices, then rrr, generates again one of the T s, the unit matrix... 

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