Cayley-Hamilton theorem states every matrix satisfies its own characteristic equation
Cayley–Hamilton theorem
Cayley-Hamilton theorem states every matrix satisfies its own characteristic equation
The Cayley-Hamilton theorem is a fundamental result in linear algebra that asserts every square matrix satisfies its own characteristic equation. This theorem provides a powerful tool for understanding the properties and behavior of matrices.
The characteristic polynomial of an n x n matrix A is defined as p_A(λ) = det(λI_n - A), where det represents the determinant operation, λ is a scalar variable, and I_n is the n x n identity matrix. This polynomial encapsulates important information about the matrix, such as its eigenvalues and eigenvectors.
Since each entry of the matrix (λI_n - A) is either constant or linear in λ, the characteristic polynomial is a polynomial in λ of degree n. According to the Cayley-Hamilton theorem, substituting the matrix A into its characteristic polynomial results in the zero matrix, demonstrating that the matrix satisfies its own characteristic equation.
Example
Consider a 2 x 2 matrix A = [[a, b], [c, d]]. The characteristic polynomial p_A(λ) = det(λI_2 - A) = det([[λ-a, -b], [c, λ-d]]) = (λ-a)(λ-d) - (-b)(c) = λ^2 - (a+d)λ + (ad-bc). According to the Cayley-Hamilton theorem, substituting A into its characteristic polynomial yields A^2 - (a+d)A + (ad-bc)I_2 = 0, showing that A satisfies its own characteristic equation.
Understanding the Cayley-Hamilton theorem is crucial for solving matrix equations and analyzing the properties of matrices in linear algebra.
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