Cayley-Hamilton

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... 

Based on the Cayley-Hamilton expression for an exponential term, one may find [14,92,93]... 

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

Because A is a 2 x 2 matrix, from Cayley-Hamilton theory [9], A can be expanded as... 

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... 

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... 

The determinant uses straight vertical bars and is a single number while a matrix has little right-angle extensions on the vertical bars and represents values in a linear system of equations. Note that a Cayley-Hamilton matrix led to a determinant due to the zeros on the right side of the secular equations. We can use these roots to find the coefficients of the 2p TT molecular orbital coefficients. When X = — 1 we can go back to the original system of equations but still use the determinant with x so we have xcj + lc2 = 0 or —cj - - C2 = 0 and ci = C2- Note that we could use either equation and we wiU get the same result However, we need to normalize the molecular orbitals so we set up the... 

Next, we combine both sides of the equation element by element to set up the Cayley-Hamilton condition. In matrix addition or subtraction [A] + [B] = [C Cj, = -E so... 

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 ... 

Thus X = 4 and 2 which are the roots of the Cayley-Hamilton polynomial as well as the eigenvalues of the matrix When X = 4 we can insert this root back into either equation to find the corresponding eigenvector. 

Although the process is for an arbitrary c, ., in effect it removes the summation over p and summations associated with the index for electron i. As with the Hiickel pi-electron derivation, this is the minimization requirement for just one (Cp,) coefficient for the orbital of just one electron (i) and just one equation in the rows of equations that form the system that leads to the Cayley— Hamilton situation. From the pi-electron case we can see that this can be put into a matrix equation where we will need to use diagonalization. 

There remains one very mysterious step. Note the subscripts on y which implies that each orbital vj ,- (which we do not know yet) is dependent somehow on all the other )>, to maintain orthogonality to them as well as normalization. Recall that the big wave function (1, 2, 3,...) is a determinant which is a single number at each (x, y, z) point in space and we know some sort of unitary transformation [T] [X,j][T ] = [X ]d,ag could be applied to the orbitals vj (if we knew them ) which would not change the value of the overall wave function, but only make the y interactions diagonal. If we assume this has been done and solve the equation subject to that constraint we will make it tmel Thus we end up with an equation for aU the electrons in a given basis set as a Cayley-Hamilton problem to find the coefficients c ,- and from them find the energy once the coefficients are known. A diagonal X eliminates the Xy for the overlap term and leads to... 

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