Homogeneous linear equation, solution

Equation (125) applies for all values of the index k — 1,2,..., m. It is a set of m simultaneous, homogeneous, linear equations for the unknown values of the coefficients c . Following Cramer s rule (Section 7.8), a nontrivial solution exists only if the determinant of the coefficients vanishes. Thus, the secular determinant takes the form... 

If the unit matrix E is of order n, Eq. (67) represents a system of n homogeneous, linear equations in n unknowns. They are usually referred to as the secular equations. According to Cramer s rule [see (iii) of Section 7.8], nontrivial solutions exist only if the determinant of the coefficients vanishes. Thus, for the solutions of physical interest,... 

Because Equation 3-lOla represents a set of homogeneous linear equations, multiplying the solution by a positive or negative factor is still a solution. Therefore, each column vector in Equation 3-lOlc and 3-lOld can be made a unit vector. Then the matrix T is obtained. With this matrix known, diffusion profiles can be calculated by solving Equation 3-99c. 

These are two homogeneous linear equations in the two unknowns a and b they have a solution only if the determinant formed by their coefficients vanishes ... 

These two equations form a system of homogeneous linear equations in c, and c2. They obviously have the trivial solutions c, = c2 = 0. It is proved in the theory of homogeneous linear equations that other, nontrivial solutions can exist only if the matrix of the coefficients of the Ci s forms a determinant equal to zero (Cramer s theorem). Thus, we have the so-called secular equation ... 

We want to find the general solution for the system of homogeneous linear equations for the linear polyene chain with N atoms yielding the Nth degree secular equation... 

Higgs, 1953a). Substitution of Eq. (42) into the above system of differential equations reduces these to a set of 3P simultaneous homogeneous linear equations in the unknowns A,. This set has a nontrivial solution only if... 

This is equivalent to solving a set of homogeneous linear equations for the Cj s. In order for the homogeneous set of equations to have a nontrivial solution, the determinant of the C, coefficients must vanish. This leads to the secular equation... 

Such a set of equations can be solved only for the ratios of the jc s i.e., any one k may be chosen and all of the others expressed in terms of it. For an arbitrary value of W kl, however, the set of equations may have no solution except the trivial one < = 0. It is only for certain values of W kl that the set of equations has non-trivial solutions the condition that must be satisfied if such a set of homogeneous linear equations is to have non-zero solutions is that the determinant of the coefficients of the unknown quantities vanish that is, that... 

For an arbitrary choice of the functions Fn(x) Equation 27-5 represents an infinite number of equations in an infinite number of unknown coefficients An. Under these circumstances questions of convergence arise which are not always easily answered. In special cases, however, only a finite number of functions Fn(x) will be needed to represent a given function p(x) in these cases we know that the set of simultaneous homogeneous linear equations 27-5 has a non-trivial solution only when the determinant of the coefficients of the A s vanishes that is, when the condition... 

These homogeneous linear equations have a nontrivial solution if the following determinant is zero ... 

This is a set of homogeneous linear equations. To obtain a nontrivial solution, the determinant of the coefficients multiplying the unknowns ci and C2 has to be zero (the secular determinant, cf., die variational method in Chapter 5) ... 

These homogeneous linear equations will have non-trivial solutions if the secular determinant is equal to zero ... 

Petho, A. (1967b). On a class of solutions of algebraic homogeneous linear equations. 

These three homogeneous linear equations for the directional cosines can only be solved for a non-zero solution, when their determinant equals zero. Thus ... 

The eigenvectors Xj are solutions of a set of homogeneous linear equations and are thus determined only up to a scalar multiplier that is, Xj and kxj will both be a solution to the equation if A is a scalar constant. We obtain a normalized eigenvector Xj when the eigenvector x, is scaled by the vector norm Xil that is... 

Thus, requiring that dEjdci vanish for all coefficients produces n homogeneous linear equations (homogeneous, all equal zero linear, all Cj s to first power). If one chooses a value for E, there remain n unknowns—the coefficients c,-. (The integrals Hij and Sij are presumably knowable since H and the functions are known.) Of course, one trivial solution for Eqs. (7-46) is always possible, namely, ci = C2 = = c = 0. But... 

Multiplying Eqs. (24) and (25) by oci and a2 and subtracting Eq. (26) using Eq. (27) we find that a and ot2 have to satisfy two homogeneous linear equations, which have a solution different from zero only when the determinant... 

Equation (G-29) represents a set of simultaneous equations, one for each value of m. The number of equations is g, and the number of unknown coefficients c j for which we can solve is g. These are homogeneous linear equations, which can be satisfied by setting each of the Cnj coefficients equal to zero. This is called the trivial solution. If we divide all of the Cnj coefficients by cn it is clear that we really have only g - I independent coefficients. We have one equation too many and the set of g equations is overdetermined, which means that unless some condition is satisfied, the trivial solution is the only solution. 

If problems persist and additional redundancies are suspected in the MCSCF wave fiuiction, these redundancies may be detected from (12.2.55) or by solution of the homogeneous linear equations (12.2.52). 

TTte use of determinants in the solution of simultaneous equations For a set of homogeneous linear equations of the form... 

The procedure we followed in the previous section was to take a pair of coupled equations, Eqs. (5-6) or (5-17) and express their solutions as a sum and difference, that is, as linear combinations. (Don t forget that the sum or difference of solutions of a linear homogeneous differential equation with constant coefficients is also a solution of the equation.) This recasts the original equations in the foiin of uncoupled equations. To show this, take the sum and difference of Eqs. (5-21),... 

It is a property of linear, homogeneous differential equations, of which the Schroedinger equation is one. that a solution multiplied by a constant is a solution and a solution added to or subtracted from a solution is also a solution. If the solutions Pi and p2 in Eq. set (6-13) were exact molecular orbitals, id v would also be exact. Orbitals p[ and p2 are not exact molecular orbitals they are exact atomic orbitals therefore. j is not exact for the ethylene molecule. 

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

If and are linearly independent solutions of the homogeneous equation (44), then its general solution can be designed as a linear combination of and 2/p with arbitrary constants Cj and Cji... 

The application of the time-independent Schrodinger equation to a system of chemical interest requires the solution of a linear second-order homogeneous differential equation of the general form... 

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