Cayley Hamilton Theorem Statement
The Cayley Hamilton Theorem says that if you have a square matrix with real or complex numbers, the special polynomial you get from that matrix, called the characteristic polynomial, will turn out to be the zero matrix. This characteristic polynomial is usually written as det(λIn−A), where det represents the determinant, λ is a variable, in is the identity matrix, and A is the given matrix.
Cayley Hamilton Theorem
This polynomial can be broken down into a simpler form, written as p(λ)=anλn+an−1λn−1+…..+a1λ+a0λ0. It’s a kind of math expression where and is the leading coefficient, which means it’s the number in front of the highest degree variable (λn), and in this case, an is always 1. The other coefficients, an−1,…,a1,a0, are the numbers in front of the other variables in decreasing order of degree, like λn−1,…,λ1,λ0.
p(A) = An + an-1 An-1 + ….. + a1A +a0In = 0
OR
p(A) = 0, where A is an n×n square matrix
Cayley Hamilton Theorem
Cayley Hamilton Theorem was introduced by mathematician Arthur Cayley. It is a crucial idea in matrix algebra. It states that each square matrix adheres to a distinct equation known as the characteristic polynomial. This polynomial, derived from adjustments to the matrix, is essential for comprehending the matrix’s characteristics.
The theorem is applied in various mathematical domains, assisting in matrix-related operations like inversion, exponentiation, and control theory. This article covers the meaning of Cayley Hamilton’s Theorem, the Statement of Cayley Hamilton’s Theorem Formula, and the Proof of Cayley Hamilton’s Theorem in 2 × 2 matrix and 3 × 3 matrix.
Table of Content
- What is Cayley Hamilton’s Theorem?
- Cayley Hamilton Theorem Statement
- Cayley Hamilton Theorem Example
- Cayley Hamilton Theorem Formula
- Cayley Hamilton Theorem for 2×2 Matrix
- Cayley Hamilton Theorem for 3×3 Matrix
- Proof of Cayley Hamilton Theorem
- Applications of Cayley Hamilton Theorem
- Examples on Cayley Hamilton Theorem