Proof of the Cayley-Hamilton Theorem

B. Schnizer

schnizer@itp.tu-graz.ac.at

2008-11-10

The Cayley-Hamilton Theorem: 
Each matrix A satisfied its own characteristic equation
Rao, § 3.14, End.

n = 3

The matrix A

n  =  3;  Clear[a]
ma  =  Array[a, {n, n}];
MatrixForm[ma]

The characteristic polynomial and its coefficients

The characteristic polynomial

chp  =  
        Det[ma  -  λ IdentityMatrix[Length[ma]]] // 
          ExpandAll;

The coefficients of the characteristic polynomial

ca  =  
        Table[ Coefficient[λ chp, λ^k], {k, 
            n + 1}];

Coefficient of  (λ^n)   equals   (-1)^n

ca[[n + 1]] ==  (-1)^n

Coefficient of  (λ^(n - 1))   equals     (-1)^nTrace(A) =  (-1)^n Sp(A)

(-1)^(n  -  1) ca[[n]]

%  ==  Tr[ma]

Coefficient of  (λ^0)   equals   (-1)^n det(A)

ca[[1]]/Det[ma] // Together

Coefficient of  (λ^(n - 2))   is (apart from the sign) the sum of the 
((n - 1) x (n - 1)) minors belonging to the main diagonal of A :

ca[[2]]

tc  =  Table[Coefficient[Det[ma], a[k, k]], {k, n}];

Apply[Plus, tc]/ ca[[2]] // Together

Each matrix A satisfies its own characteristic equation

Sum[ca[[k + 1]] MatrixPower[ma, k], {k, 0, n}] // ExpandAll // 
      MatrixForm