Cayley’s theorem

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. 

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. 

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... 

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