Cayley-Hamilton Theorem Calculator

Historical Background

The Cayley-Hamilton Theorem is a fundamental result in linear algebra, discovered independently by mathematicians Arthur Cayley and William Hamilton in the 19th century. It states that every square matrix satisfies its own characteristic equation. This theorem is crucial in understanding matrix theory and has practical applications in areas such as systems of linear equations, control theory, and quantum mechanics.

Calculation Formula

The Cayley-Hamilton theorem asserts that for a matrix \( A \), the characteristic polynomial \( p_A(\lambda) \) is given by:

\[ p_A(\lambda) = \det(\lambda I - A) \]

Where \( I \) is the identity matrix, and \( \det \) denotes the determinant. The matrix satisfies the equation:

\[ p_A(A) = 0 \]

Example Calculation

Given a 2x2 matrix:
\[ A = \begin{pmatrix} 2 & 1 \ 1 & 3 \end{pmatrix} \]

The characteristic polynomial is:
\[ \det(\lambda I - A) = \det \begin{pmatrix} \lambda - 2 & -1 \ -1 & \lambda - 3 \end{pmatrix} = (\lambda - 2)(\lambda - 3) - (-1)(-1) = \lambda^2 - 5\lambda + 6 \]

The matrix satisfies the Cayley-Hamilton theorem, as substituting \( A \) into its characteristic polynomial yields the zero matrix.

Importance and Usage Scenarios

The Cayley-Hamilton theorem simplifies computations related to matrix functions, matrix powers, and solving systems of linear equations. It has applications in linear differential equations, control theory, and quantum mechanics, where matrix exponentials and similar functions are used.

Common FAQs

1. What is the characteristic polynomial?

   • The characteristic polynomial of a matrix is a polynomial whose roots are the eigenvalues of the matrix. It is computed as \( \det(\lambda I - A) \).

2. How does the Cayley-Hamilton theorem help in practical applications?

   • It allows us to express higher powers of a matrix in terms of lower powers, simplifying calculations in linear algebra, control systems, and quantum mechanics.

3. Does the theorem apply to all matrices?

   • Yes, the Cayley-Hamilton theorem applies to all square matrices, regardless of their size or the field over which they are defined.