Matrix 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. 

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... 

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. 

This is a linear system of equations in n variables where the unknown variables are the c, coefficients. Formally, this calls into effect the Cayley-Hamilton theorem [6] because the right-hand sides of the equations are all zero. The Cayley-Hamilton theorem [6] states that a square matrix. A, satisfies its characteristic equation and if we have a characteristic polynomial of the eigenvalues of the matrix... 

Since the right side constants are 0, we have to invoke the Cayley-Hamilton theorem just as we did with the pi-electron treatment of ethylene. We force the determinant of the coefficients to be zero Then the eigenvalues of the matrix will be the roots of the polynomial in X, which fulfills the Cayley-Hamilton Theorem ... 

The well-known Cayley-Hamilton theorem states that a. square matrix satisfies its own... 

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