Characteristic polynomial

Since benzenoid graphs are bipartite, their characteristic polynomials can be written in the form 

Special methods for the calculation of the characteristic polynomials of benzenoid graphs have recently been designed by Sachs and John [16, 17]. Their method is especially efficient in the case of catacondensed systems. 

Explicit combinatorial expressions are known for the first few coefficients of (f (B, x) [2, 21, 33, 39, 40]. We will skip these results because in the subsequent paragraph the spectral moments are discussed at due length. Using the Newton identities [24] it is easy to compute the coefficients of the characteristic polynomial from spectral moments and vice versa. 

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A square matrix has the eigenvalue A if there is a vector x fulfilling the equation Ax = Ax. The result of this equation is that indefinite numbers of vectors could be multiplied with any constants. Anyway, to calculate the eigenvalues and the eigenvectors of a matrix, the characteristic polynomial can be used. Therefore (A - AE)x = 0 characterizes the determinant (A - AE) with the identity matrix E (i.e., the X matrix). Solutions can be obtained when this determinant is set to zero. 

The inside terms of the determinant represent the characteristic polynomial P(A) of A ... 

Some structural information about G can be obtained directly from the characteristic polynomial of A G) ... 

Although the actual cycle decomposition (as well as the tree structure) of a particular graph is determined exactly by the set of elementary divisors i(a ), much of the general form of the possible dynamics may be extracted from Pl x) itself. All graphs whf)se characteristic polynomials Pii=P Yi=i Pi AY (mod q), for. some fixed P ( / ), for example, mu.st share the following properties ... 

John, P. and Sachs, H. Calculating the Numbers of Perfect Matchings and of Spanning Tress, Pauling s Bond Orders, the Characteristic Polynomial, and the Eigenvectors of a Benzenoid System. 153, 145-180 (1990). 

Before this is done, however, a certain paradox needs to be discussed briefly. Given a matrix A, and a nonsingular matrix V, it is known that A, and V XA V, have the same characteristic polynomial, and the two matrices are said to be similar. Among all matrices similar to a given matrix A, there are matrices of the form... 

The methods of simple and of inverse iteration apply to arbitrary matrices, but many steps may be required to obtain sufficiently good convergence. It is, therefore, desirable to replace A, if possible, by a matrix that is similar (having the same roots) but having as many zeros as are reasonably obtainable in order that each step of the iteration require as few computations as possible. At the extreme, the characteristic polynomial itself could be obtained, but this is not necessarily advisable. The nature of the disadvantage can perhaps be made understandable from the following observation in the case of a full matrix, having no null elements, the n roots are functions of the n2 elements. They are also functions of the n coefficients of the characteristic equation, and cannot be expressed as functions of a smaller number of variables. It is to be expected, therefore, that they... 

But then is the characteristic polynomial of A, and its coefficients are the elements of / and can be found by solving Eq. (2-11). This is essentially the method of Krylov, who chose, in particular, a vector et (usually ej) for vx. Several methods of reduction of the matrix A can be derived from applying particular methods of inverting or factoring V at the same time that the successive columns of V are being developed. Note first that if... 

Theobald, D. L. Rapid calculation of RMSDs using a quaternion-based characteristic polynomial. Acta Crystallogr. 2005, A61, 478 80. 

The time-response characteristics of a model can be inferred from the poles, i.e., the roots of the characteristic polynomial. This observation is independent of the input function and singularly the most important point that we must master before moving... 

In addition, the time dependence of the solution, meaning the exponential function, arises from the left hand side of Eq. (2-2), the linear differential operator. In fact, we may recall that the left hand side of (2-2) gives rise to the so-called characteristic equation (or characteristic polynomial). 

The time dependence of the time domain solution is derived entirely from the roots of the polynomial in the denominator (what we will refer to later as the poles). The polynomial in the numerator affects only the coefficients a. This is one reason why we make qualitative assessment of the dynamic response characteristics entirely based on the poles of the characteristic polynomial. 

The inherent dynamic properties of a model are embedded in the characteristic polynomial of the differential equation. More specifically, the dynamics is related to the roots of the characteristic polynomial. In Eq. (2-27), the characteristic equation is xs + 1 = 0, and its root is -1/x. In a general sense, that is without specifying what C in is and without actually solving for C (t), we... 

We know that G(s) contains information on the dynamic behavior of a model as represented by the differential equation. We also know that the denominator of G(s) is the characteristic polynomial of the differential equation. The roots of the characteristic equation, P(s) = 0 pb p2,... pn, are the poles of G(s). When the poles are real and negative, we also use the time constant notation ... 

We now put one and one together. The key is that we can "read" the poles—telling what the form of the time-domain function is. We should have a pretty good idea from our exercises in partial fractions. Here, we provide the results one more time in general notation. Suppose we have taken a characteristic polynomial, found its roots and completed the partial fraction expansion, this is what we expect in the time-domain for each of the terms ... 

In this rearrangement, xp is the process time constant, and Kd and Kp are the steady state gains.2 The denominators of the transfer functions are identical, they both are from the LHS of the differential equation—the characteristic polynomial that governs the inherent dynamic characteristic of the process. 

The important observation is that when we "close" a negative feedback loop, the numerator is consisted of the product of all the transfer functions along the forward path. The denominator is 1 plus the product of all the transfer functions in the entire feedback loop ( .e., both forward and feedback paths). The denominator is also the characteristic polynomial of the closed-loop system. If we have positive feedback, the sign in the denominator is minus. 

Take note (again ) that the characteristic polynomials in the denominators of both transfer functions are identical. The roots of the characteristic polynomial (the poles) are independent of the inputs. It is obvious since they come from the same differential equation (same process or system). The poles tell us what the time-domain solution, y(t), generally would "look" like. A final reminder no matter how high the order of n may be in Eq. (3-4), we can always use partial fractions to break up the transfer functions into first and second order terms. 

This is a form that serves many purposes. The term in the denominator introduces a negative pole in the left-hand plane, and thus probable dynamic effects to the characteristic polynomial of a problem. The numerator introduces a positive zero in the right-hand plane, which is needed to make a problem to become unstable. (This point will become clear when we cover Chapter 7.) Finally, the approximation is more accurate than a first order Taylor series expansion.1... 

We do not need to carry the algebra further. The points that we want to make are clear. First, even the first vessel has a second order transfer function it arises from the interaction with the second tank. Second, if we expand Eq. (3-46), we should see that the interaction introduces an extra term in the characteristic polynomial, but the poles should remain real and negative.1 That is, the tank responses remain overdamped. Finally, we may be afraid( ) that the algebra might become hopelessly tangled with more complex models. Indeed, we d prefer to use state space representation based on Eqs. (3-41) and (3-42). After Chapters 4 and 9, you can try this problem in Homework Problem 11.39. 

One important observation that we should make immediately the characteristic polynomial of the matrix A (E4-7) is identical to that of the transfer function (E4-2). Needless to say that the eigenvalues of A are the poles of the transfer function. It is a reassuring thought that different mathematical techniques provide the same information. It should come as no surprise if we remember our linear algebra. 

Comments at the end of Example 4.1 also apply here. The result should be correct, and we should find that both the roots of the characteristic polynomial p and the eigenvalues of the matrix a are -0.2 0.98j. We can also check by going backward ... 

The returned vector p is obviously the characteristic polynomial. The matrix ql is really the first column of the transfer function matrix in Eq. (E4-30), denoting the two terms describing the effects of changes in C0 on Ci and Cj Similarly, the second column of the transfer function matrix in (E4-30) is associated with changes in the second input Q, and can be obtained with ... 

If A has repeated eigenvalues (multiple roots of the characteristic polynomial), the result, again from introductory linear algebra, is the Jordan canonical form. Briefly, the transformation matrix P now needs a set of generalized eigenvectors, and the transformed matrix J = P 1 AP is made of Jordan blocks for each of the repeated eigenvalues. For example, if matrix A has three repealed eigenvalues A,j, the transformed matrix should appear as... 

For review after the chapter on root locus with the strategy in Fig. 5.3, the closed-loop characteristic polynomial and thus the poles remain the same, but not the zeros. You may also... 

Closed-loop transfer functions and characteristic polynomials... 

The important point is that the dynamics and stability of the system are governed by the closed-loop characteristic polynomial ... 

In real life, we expect probable simultaneous reference and disturbance inputs. As far as analysis goes, the mathematics is much simpler if we consider one case at a time. In addition, either case shares the same closed-loop characteristic polynomial. Hence they should also share the same stability and dynamic response characteristics. Later when we talk about integral error criteria in controller design, there are minor differences, but not sufficient to justify analyzing a problem with simultaneous reference and load inputs. 

Reminder the characteristic polynomial is the same in feedback system, either case. 

With the given choices of Gc (P, PI, PD, or PID), Gp, Ga and Gm, plug their transfer functions into the closed-loop equation. The characteristic polynomial should fall out nicely. 

The key is to recognize that the system may exhibit underdamped behavior even though the open-loop process is overdamped. The closed-loop characteristic polynomial can have either real or complex roots, depending on our choice of Kc. (This is much easier to see when we work with... 

You should be able to fill in the gaps and finish the rest of the work in deriving the transfer functions. In this case, we may want to use the steam mass flow rate as the manipulated variable. The transfer function relating its effect on T will be second order, and the characteristic polynomial does not have the clean form in simpler textbook examples. 

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