Cayley theorem

The solution of Eq. (42) directly follows the Hamilton-Cayley theorem, the result being... 

Ilamilton-Cayley Theorem. A second method of determining the coefneients Cp is based upon the following theorem If the characteristic ( (luation of H is... 

CayA89 Cayley, A. A theorem on trees. Quar. J. Pure Appl. Math. 23 (1889) 376-378. 

The Linear Algebraic Problem.—Familiarity with the basic theory of finite vectors and matrices—the notions of rank and linear dependence, the Cayley-Hamilton theorem, the Jordan normal form, orthogonality, and related principles—will be presupposed. In this section and the next, matrices will generally be represented by capital letters, column vectors by lower case English letters, scalars, except for indices and dimensions, by lower case Greek letters. The vectors a,b,x,y,..., will have elements au f it gt, r) . .. the matrices A, B,...,... 

Roughly, the Ackermann s formula arises from the application of the Cayley-Hamilton theorem... 

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. 

Remark f. Notice that for a linear system the coefficients ao,, ai j,..., in equation (48) represent the coefficients of the characteristic equation of matrix S. For the nonlinear case, these coefficients do not represent a generalization of the Cayley-Hamilton theorem hence the assumption is necessary for the existence of the solution of the NRRP. 

In the last line we have made use of the Cayley-Hamilton theorem. The... 

The recursion (84) can be extended to operators and matrices. This is done by using the Cayley-Hamilton theorem [2], which states that for a given analytic scalar function/(m), the expression for its operator counterpart/(U) is obtained via replacement of u by U as in Eq. (6). In this way, we can introduce the Lanczos operator and matrix polynomials defined by the following recursions ... 

The Cayley-Hamilton theorem is one of the most powerful theorems of matrix theory. It states A matrix satisfies its own characteristic equation. That is, if the characteristic equation of an m X m matrix [A] is... 

The enumeration of constitutional isomers by Cayley, then by Henze and Blair, and finally by means of Polya s theorem or double cosets was reviewed earlier. 

This relation can be reduced, by using the Cayley-Hamilton s theorem. 

In his Memoir on the Theory of Matrices, Cayley mentioned the important theorem for matrices, known as the Cayley-Hamilton theorem, which states that a square matrix satisfies its own characteristic polynomial. The significance of the Cayley-Hamilton theorem is that for a matrix of size n x n all information is in the first A" matrices, n = 1,... n. Thus, there is no new information to be obtained by calculating higher powers of matrices. 

Note 2.6 (Cayley-Hamilton Theorem). The well-known Cayley-Hamilton theorem states that... 

---