Cayley-Hamilton Theorem: Statement & Proof, Examples - JEE PDF Download

Introduction

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In the realm of linear algebra, the Cayley-Hamilton theorem, named after mathematicians Arthur Cayley and William Rowan Hamilton, establishes that every square matrix belonging to a commutative ring (such as the real or complex field) satisfies its own characteristic equation. If we have a given n×n matrix A and In represents the n×n identity matrix, the unique polynomial of A can be expressed as:
p(x) = det(xI_n – A)
Here, the determinant operation is denoted by 'det', and x represents the scalar element of the base ring. Since the entries of the matrix are polynomials in x (either linear or constant), the determinant itself becomes an nth-order monic polynomial in x.

What is Cayley–Hamilton theorem?

The Cayley-Hamilton theorem can be restated as follows: When we substitute the matrix A for x in the polynomial p(x) = det(xI_n - A), the result is the zero matrix, expressed as

p(A) = 0.

This theorem asserts that an n x n matrix A is annihilated by its characteristic polynomial, det(tI - A), which is a monic polynomial of degree n. The powers of A, obtained by substituting powers of x, are determined through repeated matrix multiplication. The constant term of p(x) yields a multiple of the power A⁰, where the power corresponds to the identity matrix. The theorem allows us to express Aⁿ as a linear combination of the lower powers of A. If the ring under consideration is a field, the Cayley-Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.

Example of Cayley-Hamilton Theorem

1. 1 x 1 Matrices

For 1 x 1 matrix A(a_1,1) the characteristic polynomial is given by
p(λ) = λ - α
So, p(A) = (α) – (a_1,1)  = 0 is obvious.
2. 2 x 2 Matrices
Let us look this through an example


The Cayley-Hamilton claims that if, we define

We can verify this result by computation

For a generic 2 x 2 matrix,

The resultant polynomial is given by:

So the Cayley-Hamilton theorem states that

it is always the case, which is evident by working out on  A².