The Cayley-Hamilton Theorem

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 many cases it is very difficult to compute functions of matrices from their definitions 

Let P[[ ]] be a matrix polynomial of arbitrary degree that we wish to compute. Then P(A) represents the corresponding polynomial of A. A theorem of algebra states that there exist polynomials (A) and r(A) such that 

The procedure outlined above for a 2 x 2 matrix may be extended to m X m matrices. This involves the solution of m simultaneous linear equations in m unknowns, 

---

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 ... 

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. 

By applying the Cayley-Hamilton theorem, the previous equation can rewritten... 

Note that D° = I and for an incompressible fluid f = —p. Again we can evoke the Cayley Hamilton theorem, eq. 1.6.2. Thus... 

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... 

That works fine and can be extended to N unknowns as long as the right sides of the equation are nonzero. In the variational molecular orbital problem, all the right sides of the equation are zero and if we use Cramer s rule we only get the trivial solution with all values equal to zero. The Cayley-Hamilton theorem implies that if all the equations equal zero, you can still get a nonzero solution by forcing the denominator determinant to be zero, that is, by solving for the roots of the corresponding... 

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 Cayley-Hamilton theorem states that if the characteristic equation of a symmetric second-order tensor A is C(X) = 0, then the tensor A satisfies the equation C(A) = 0... 

The Cayley-Hamilton theorem has been widely used in the formulation of constitutive equations (Rivlin and Ericksen 1955). As an example, let the stress tensor a be expressed as a function of powers of the rate-of-deformation tensor d ... 

Applying the Cayley-Hamilton theorem [18,20,28], equation (4.3) can be reduced to the form... 

---