Caley-Hamilton Theorem

 

The Caley-Hamilton Theorem is a fundamental result in linear algebra that states that every square matrix satisfies its own characteristic polynomial.

The characteristic polynomial of a square matrix A is a polynomial equation of the form |A - λI| = 0, where λ is a scalar value and I is the identity matrix. The roots of this equation, known as the eigenvalues of the matrix, are the values of λ that make the equation true.

The Caley-Hamilton Theorem states that for any nxn matrix A, there exists a polynomial p(λ) of degree n such that p(A) = 0. This means that the matrix A is a root of its own characteristic polynomial.

For example, consider the following 2x2 matrix:

[2 3] [1 2]

The characteristic polynomial of this matrix is:

|[2 3] - λ[1 0]| = 0 |[1 2] [0 1]|

This simplifies to:

(2 - λ)(2 - λ) - (3)(1) = 0

Which gives us the characteristic polynomial:

λ^2 - 4λ + 1 = 0

According to the Caley-Hamilton Theorem, this polynomial should be equal to the matrix [2 3] multiplied by a scalar, plus the matrix [1 2] multiplied by another scalar, plus the scalar -1. This can be verified by performing the multiplication:

[2 3] * λ^2 + [1 2] * (-4) + (-1) = [2 3] * λ^2 - [1 2] * 4 - 1 = [4 6] - [4 8] - 1 = [0 -2] - 1 = [-1 -2]

Therefore, the Caley-Hamilton Theorem holds for this matrix.