Maths - Cayley-Hamilton Theorem

This theorem says that every square matrix (over a field or a commutative ring) satisfies its own characteristic equation. That is:

if: p(λ) = det(λ I - A)            characteristic polynomial of A
then : p(A) = 0    

where:

An equivalent statement to Cayley-Hamilton is that the minimal polynomial of a square matrix divides its characteristic polynomial.

Examples

I have used a computer algebra programs called Axiom for these examples, how to install Axiom here.

I have put user input in red:

a1 := matrix[[1,0],[0,1]]

        +1  0+
   (1)  |    |
        +0  1+
                                                    Type: Matrix(Integer)
m1 := diagonalMatrix[lambda,lambda]-a1

        +lambda - 1      0     +
   (2)  |                      |
        +    0       lambda - 1+
                                        Type: Matrix(Polynomial(Integer))
d1 := determinant(m1)

              2
   (3)  lambda  - 2lambda + 1
                                                Type: Polynomial(Integer)
a1*a1 -2*a1 + diagonalMatrix[1,1]

        +0  0+
   (4)  |    |
        +0  0+
                                                    Type: Matrix(Integer)
a2 := matrix[[a,b],[c,d]]
              
              
        +a  b+
   (5)  |    |
        +c  d+
                                        Type: Matrix(Polynomial(Integer))
m2 := diagonalMatrix[lambda,lambda]-a2
              
              
        +lambda - a     - b    +
   (6)  |                      |
        +   - c      lambda - d+
                                        Type: Matrix(Polynomial(Integer))
d2 := determinant(m2)

              2          
   (7)  lambda  + (- d - a)lambda + a d - b c
                                                Type: Polynomial(Integer)
a2*a2+(-d-a)*a2 + determinant(a2)*diagonalMatrix[1,1]

        +0  0+
   (8)  |    |
        +0  0+
                                        Type: Matrix(Polynomial(Integer))
a3 := matrix[[1,0,0],[0,1,0],[0,0,1]]

        +1  0  0+
        |       |
   (9)  |0  1  0|
        |       |
        +0  0  1+
                                                    Type: Matrix(Integer)
m3 := diagonalMatrix[lambda,lambda,lambda]-a3


         +lambda - 1      0           0     +
         |                                  |
   (10)  |    0       lambda - 1      0     |
         |                                  |
         +    0           0       lambda - 1+
                                        Type: Matrix(Polynomial(Integer))
d3 := determinant(m3)


               3          2
   (11)  lambda  - 3lambda  + 3lambda - 1
                                                Type: Polynomial(Integer)
a3*a3*a3 - 3*a3*a3 +3*a3 - diagonalMatrix[1,1,1]


         +0  0  0+
         |       |
   (12)  |0  0  0|
         |       |
         +0  0  0+
                                                    Type: Matrix(Integer)
a4 := matrix[[a,b,c],[d,e,f],[g,h,i]]


         +a  b  c+
         |       |
   (13)  |d  e  f|
         |       |
         +g  h  i+
                                        Type: Matrix(Polynomial(Integer))
m4 := diagonalMatrix[lambda,lambda,lambda]-a4


         +lambda - a     - b         - c    +
         |                                  |
   (14)  |   - d      lambda - e     - f    |
         |                                  |
         +   - g         - h      lambda - i+
                                        Type: Matrix(Polynomial(Integer))
d4 := determinant(m4)


   (15)
       3                      2
 lambda  + (- i - e - a)lambda  + ((e + a)i - f h - c g + a e - b d)lambda

 (- a e + b d)i + (a f - c d)h + (- b f + c e)g
                                            Type: Polynomial(Integer)
a4*a4*a4 + (-i-e-a)*a4*a4
+((e + a)*i - f*h - c*g + a*e - b*d)*a4 
+ ((- a*e + b*d)*i + (a*f - c*d)*h + (- b*f + c*e)*g)*diagonalMatrix[1,1,1]


         +0  0  0+
         |       |
   (16)  |0  0  0|
         |       |
         +0  0  0+
                                        Type: Matrix(Polynomial(Integer))
(17) ->

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