Matrix characteristic polynomial

Spialter, L. (1964b). The Atom Connectivity Matrix Characteristic Polynomial (ACMCP) and its Physico-Geometric (Topological) Significance. J.Chem.Doc., 4, 269-274. 

L. Spialter, The atom connectivity matrix characteristic polynomial (ACMCP) and its physico-geometric (topological) significance, J. Chem. Doc. 4 (1964b) 269-274. 

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. 

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... 

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... 

At this point, since S is q x q matrix, according to the Cayley-Hamilton Theorem, S must satisfy its own characteristic polynomial, i.e. 

The characteristic polynomial may be obtained readily from the adjacency matrix A by placing x s on the main diagonal and expanding the determinant (39). 

Analogously to the characteristic polynomial derived from adjacency matrix A, Hosoya et al. 64> introduced the distance polynomial ... 

The reader may wish to compare this Spectral Theorem to Proposition 4.4. Proof. To find the eigenvalues of A, we consider its characteristic polynomial. Then we use eigenvectors to construct the matrix M. 

Note that the output of roots(p) and eig(compan(p)) each is a complex column vector of length three, i.e., each output lies in C3 and the two solution vectors are identical. Our trial polynomial p(x) = a 3 — 2a 2 + 4 has one pair of complex conjugate roots 1.5652 1.0434 i and one real root -1.1304. The (first row) companion matrix P of a normalized nth degree polynomial p (normalized, so that the coefficient an of xn in p is 1) is the sparse n by n matrix P = C(p) as described in formula (1.2). Note that our chosen p is normalized and has zero as its coefficient ai for a = a 1, i.e., the (1, 2) entry in P is zero. For readers familiar with determinants and Laplace expansion, it should be clear that expanding det(P — xI) along row 1 establishes that our polynomial p(x) is the characteristic polynomial of P. Hence P s eigenvalues are precisely the roots of the given polynomial p. 

Expansion of the Hiickel orbital (HMO) secular determinant for a PAH graph gives the characteristic polynomial P(G X) = det X1 — A where I is the identity matrix and A is the adjacency matrix for the corresponding graph [11]. The characteristic polynomial of a N carbon atom system has the following form... 

Note that a characteristic polynomial of the square matrix A = o of the order n is called a determinant for a set of linear homogeneous equations... 

Calculation of the coefficients dt for a given matrix is a very laborious process. We will give a method to calculate these coefficients proceeding directly from the complex reaction graph. Like a steady-state kinetic equation, a characteristic polynomial will be represented in the general (struc-turalized) form ... 

Theorem 1 A simple transformation of the characteristic polynomial of such a matrix will present it in a form where the contribution from each order of permutation to the value of its determinant is displayed as an elementary symmetric function of the eigenvalues of S — I. 

Moreover, for a (M M) transfer matrix T with characteristic polynomial... 

The first of these conditions expresses the diagonalization of the matrix Kh and not that of h. To satisfy the relation (1.16) the eigenvalues of KA must consist of pairs whose sums are zero (the characteristic polynomial of KA must be in z2). 

This recurrence relation and the available triple set an, fin uk] are sufficient to completely determine the state vector Q without any diagonalization of the associated Jacobi matrix U = c0J, which is given in Eq. (60). Of course, diagonalization of J might be used to obtain the eigenvalues uk, but this is not the only approach at our disposal. Alternatively, the same set uk is also obtainable by rooting the characteristic polynomial or eigenpolynomial from... 

The roots uk f=1 of Eq. (103) coincide with the eigenvalues of matrix U. On the other hand, the function det[wl — U] is proportional to its Kth degree characteristic polynomial, which is equal to QK(u) up to an overall multiplicative constant of the normalization type ... 

The characteristic polynomial of G is just the characteristic polynomial of the adjacency matrix. It will be denoted by cp(G). Then... 

Spialter L (1964) The atom connectivity matrix (ACM) and its characteristic polynomial (ACMCP), J Chem Doc 4 261... 

Corollary two the sum of the roots of a characteristic polynomial, that is, the sum of the eigenvalues of the topological matrix of a molecule, is equal to zerort ... 

The system of differential equations (E3.13) can be reduced to a single differential equation for the determinant of J h ) through the application of Davidenko s (1960) trace theorem let (s) be an n x matrix whose elements are differentiable functions in s. Let (s) be the characteristic polynomial associated with (s). Then, for all s for which (s) 0,... 

The eigenvalues X of the matrix M are the n roots of its characteristic polynomial and the set of the eigenvalues is called spectrum of a matrix A(M). Determinant and trace of M are given by the following expressions ... 

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