Linear equations homogeneous

If a material system experiences a continuous action, or a complete cycle of operations, of a perfectly reversible kind, the quantities of heat which it takes in at different temperatures are subject to a homogeneous linear equation, of which the coefficients are the reciprocals of these temperatures. If Qr be... 

Now, we can solve the system of homogeneous linear equations for the unknowns and v 22 which are the non-normalized elements of v,2 and V22-... 

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,... 

The remaining part of the mode seeking procedure can formally proceed exactly in the same as for the straight waveguide starting from both innermost and outermost slices, two values of the immittance matrix in some suitably chosen radial position p are found and are then used to construct the set of homogeneous linear equations for the mode field amplitudes ... 

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 ... 

Given this form of molecular orbitals, the Fock equations yield a system of homogeneous linear equations in the c/s, the Roothaan equations77... 

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... 

Homogeneous Linear Equations with Constant Coefficients... 

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... 

The concept of linear dependence is important in the study of homogeneous linear equations. A pair of homogeneous linear equations is similar to those of Eq. 

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

Once the complex Floquet eigenenergy has been determined from an iter-afive resolution of the implicit energy-dependent Eq. (53), the multichannel wavefunction written as a column vector T (R) can be calculated at the matching point because it satisfies the set of homogeneous linear equations ... 

How to balance chemical equations was always a continuing field because of its importance [2-6]. A critical review has been given by Herndon [7]. A general procedure to obtain the stoichiometric coefficient is to solve the system of the homogeneous linear equations that are obtained from the principles of conservation of matter and charge. The earliest chemical paper by J. Bottonily in 1878 used this method [8]. 

---