Cayley Hamilton Theorem

Cayley–Hamilton theorem says that each square matrix over a commutative ring (including the real or complex field) agrees with its equation. If A is assumed as n×n matrix and In is the identity matrix, then the distinctive polynomial of A is expressed as

P(x) = det (xln - A)

Where det stands for determinant operation and for the scalar element of the base ring, the variable would be considered as x. As the inputs of the matrix are (linear or polynomial) in x, the det or determinant is also considered as an nth order monic polynomial in x.

A polynomial m(x) is therefore said to be the minimal polynomial of A if (i) m(A) = 0; (ii) m(x) is a monic polynomial (the coefficient of the highest degree term is 1); (iii) if a polynomial g(x) is such that g(A) = 0, then m(x) divides g(x).

History of Cayley-Hamilton Theorem

Hamilton Theorem was proved in the year 1855 in terms of a linear function of the quaternion, a non-commutative ring by Cayley-Hamilton. This theorem persists for general quaternion matrices. This complies with the specific illustration of certain real 4x4 matrices over 2x2 complex matrices. Cayley–Hamilton stated Hamilton Theorem for 3x3 small matrices, but able to publish a proof for the 2x2 case. The general case was initially verified by Frobenius in 1878.

Computation of the Matrix Exponential e At The matrix exponential is simply one case of an analytic function as described above. e At = nX−1 k=0 αkAk (5) where the αi ’s are determined from the set of equations given by the eigenvalues of A. e λit = nX−1 k=0 αkλ k i

Every square matrix satisfies its characteristic equation, that is, if f(x) is the characteristic polynomial of a square matrix A, then f(A) = 0.

A polynomial m(x) is said to be the minimal polynomial of A if (a) m(A) = 0; (b) m(x) is a monic polynomial ( i.e, the coefficient of the highest degree term is 1); (c) if a polynomial g(x) is such that g(A) = 0, then m(x) divides g(x). 

The minimal polynomial of a matrix is unique. The minimal polynomial divides its characteristic polynomial.

The minimal polynomial and the characteristic polynomial have the same roots. Here is the proof : If we let f(x) and m(x) be the characteristic and minimal polynomial of a matrix respectively, then f(x) = g(x)m(x). If α is a root of m(x), then it is also a root of f(x). Simultaneously, if α is a root of f(x), then α is an eigenvalue of the matrix. Thus, there is also  a non-zero eigenvector v such that Av = αv, this implies m(A)v = m(α)v, i.e., m(α)v = 0, and v 6= 0 so that m(α) = 0.

Similar matrices have the same minimal polynomials. Here is the proof : If we let A and B be two similar matrices, then A = P −1BP for some invertible matrix P. Let m1(x) = a0 + a1x + . . . + x n and m2(x) = b0 + b1x + . . . + x l be the respective minimal polynomials of A and B. Then m2(A) = 0, which implies m1(x)|m2(x). Similarly m1(B) = 0, which implies m2(x)|m1(x). 

Let A ∈ Mn(F) and λ1, λ2, . . . , λk ∈ F be all eigenvalues of A, where λi 6= λj

for i 6= j. The matrix A is diagonalizable if and only if its minimal polynomial is a product of distinct linear polynomials,

that is, m(x) = (x − λ1)(x − λ2)· · ·(x − λk), where λi’s are distinct elements of F.

A matrix A ∈ Mn(R) such that A2 − 3A + 2I = 0 is diagonalizable. Let’s take g(x) = x 2 − 3x + 2, then g(A) = 0. Note that g(x) = (x − 1)(x − 2) and the minimal polynomial m(x) of A divides g(x). Therefore, either m(x) = (x − 1) or m(x) = (x − 2) or m(x) = (x − 1)(x − 2). In either case, the minimal polynomial is a product of distinct linear polynomials, hence diagonalizable.

Hamilton Theorem Proof

The Hamilton theorem states that if matrices A will be replaced instead of x in polynomial, p (x) = det (xln- A), it will give away the zero matrices, such as

P (A) = 0

The theory states that the nxn matrix is eradicated by its characteristic polynomial det (ti- a) which is monic of degree n. The power of A is found by replacing the power of x, which is identified by recurrent matrix multiplication, the constant term of p(x) yields a multiple of the power A₀, which power is indicated as identity matrix. This theorem permits Aᶯ to be considered as a linear combination of matrix power of A. If the ring is assumed as a field, the Cayley–Hamilton theorem would be equal to the statement which states that the smallest polynomial of a square matrix will be divided by its characteristic polynomial.

Computation of the Matrix Exponential e At The matrix exponential is simply one case of an analytic function as described above. e At = nX−1 k=0 αkAk (5) where the αi ’s are determined from the set of equations given by the eight values of A. e λit = NX−1 k=0 αkλ k i.

Cayley-Hamilton Theorem Example

Here you can see, Cayley-Hamilton theorem example based on Cayley-Hamilton Theorem proof which is stated above.

Example- If A is 3×3 matrix, then is characteristics equation would be considered as

=  │A-λI│ = 0

= λ³ + C1 λ2 + C2λ + C3I

= Substituting λ with A

= A³ +C1A2 + C2A + C3I

Explanation –

Let us consider A as n×n as a square matrix, then its characteristics polynomial will be stated as:

P (λ) =│A-λ In │ = 0

In = Identity matrix similar order as A

According to the Hamilton theorem:

P (A) = 0

Here 0, signifies the zero matrices of same order A

Cayley-Hamilton Theorem Solved Example - Will be uploaded soon.

Quiz Time

1. The Cayley-Hamilton theorem proof deals only with

a) Inverse Matrix

b) Square Matrix

c) Identity Matrix

d) Orthogonal Matrix

2. The number of rows and columns in the square matrix is

a) More than 10

b) Similar

c) Multiples of each other

d) Different

3.  Find characteristic polynomials, if the 2×2 matrix has 2 and 3 in the first row, 0 and 1 in the second row.

a) λ² - 3 λ + 2

b) λ² + 2

c) λ² + 3

d) λ² - 2 λ + 3

Fun Facts

• The Cayley-Hamilton theorem was initially proved in the year 1853, in the form of the inverse of linear equation by a quaternion, a non -commutative ring through Hamilton

• The result of the theory was first verified by Frobenius in the year 1878.

• The first record of the Cayley-Hamilton theorem was accidentally created by William Rowan Hamilton in his book “Lectures on Quaternions”.

• Arthur Cayley in 1858 applied Hamilton theory in the world of the matrix.

• The expected results can be seen when the same theory is applied by Cayley in the matrix of size 3×3. Thus the Cayley-Hamilton theorem was discovered.