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:
- det = determinant
- A = square matrix
- λ = scalar value
- I = identity matrix
- p() = polynomial
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) ->