Cayley-Hamilton Theorem Calculator
States that every square matrix satisfies its own characteristic equation.
Formula first
Overview
The Cayley-Hamilton Theorem asserts that every square matrix satisfies its own characteristic equation, meaning if p(λ) is the characteristic polynomial of matrix A, then p(A) results in the zero matrix. This fundamental result bridges the gap between matrix algebra and polynomial theory, providing a powerful tool for matrix analysis.
Apply it well
When To Use
When to use: Apply this theorem when calculating large powers of a matrix or finding the inverse of a non-singular matrix without row reduction. It is also used to simplify matrix-valued functions and to find the minimal polynomial of a linear operator.
Why it matters: It drastically reduces computational complexity in fields like control theory and signal processing by converting matrix exponentiation into linear combinations of lower powers. It is a cornerstone of the Jordan Canonical Form and other structural decompositions in linear algebra.
Avoid these traps
Common Mistakes
- Applying the theorem to non-square matrices.
- Forgetting to multiply the constant term by the identity matrix when evaluating p(A).
One free problem
Practice Problem
Given a 2×2 matrix A with diagonal elements m11 = 5 and m22 = 3, the Cayley-Hamilton theorem states that A satisfies the equation A² - kA + dI = 0. Find the value of k, which corresponds to the trace of the matrix.
Solve for:
Hint: The trace of a matrix is the sum of its diagonal elements and appears as the negative coefficient of the λ term in the characteristic polynomial.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Wikipedia: Cayley-Hamilton theorem
- Linear Algebra and Its Applications (5th ed.) by David C. Lay
- Introduction to Linear Algebra (5th ed.) by Gilbert Strang
- Linear Algebra and Its Applications by David C. Lay
- Introduction to Linear Algebra by Gilbert Strang
- Linear Algebra and Its Applications, David C. Lay