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Home›Matrix Tools›Matrix Characteristic Polynomial Calculator

Characteristic Polynomial Calculator

Compute det(A - λI) for matrices

Enter 2x2 Matrix A

Characteristic Polynomial Formula

p(λ) = det(A - λI)

2x2: p(λ) = λ² - tr(A)λ + det(A)

3x3: p(λ) = λ³ - tr(A)λ² + tr2(A)λ - det(A)

tr2(A) = (tr(A)² - tr(A²))/2

The characteristic polynomial p(λ) = det(A - λI) has eigenvalues as its roots. By Cayley-Hamilton theorem, p(A) = 0.

⚠ For n×n matrix: p(λ) = Σ_{k=0}^{n} (-1)^k * e_k * λ^(n-k) where e_k are elementary symmetric sums.

What is Characteristic Polynomial?

The characteristic polynomial of a matrix A is defined as p(λ) = det(A - λI), where I is the identity matrix. Its roots are the eigenvalues of A. This polynomial plays a fundamental role in linear algebra.

Eigenvalues

Roots of characteristic polynomial are eigenvalues of A.

Cayley-Hamilton

p(A) = 0: matrix satisfies its own characteristic equation.

Trace Relation

Sum of eigenvalues = trace(A). Product = det(A).

Monic Polynomial

Leading coefficient is (-1)^n for n×n matrix.

Teaching Example: A = [[1,2],[3,4]]
1. p(λ) = det([[1-λ,2],[3,4-λ]])
2. p(λ) = (1-λ)(4-λ) - 2*3
3. p(λ) = λ² - 5λ - 2
4. Eigenvalues: λ = (5 ± √33)/2

Applications

Eigenvalue Computation Cayley-Hamilton Matrix Powers Jordan Form Linear Algebra

Frequently Asked Questions

What is characteristic polynomial?▼

p(λ) = det(A - λI). Roots are eigenvalues of A. Used in Cayley-Hamilton theorem.

How to compute it?▼

Calculate determinant of (A - λI). For n×n: p(λ) = Σ(-1)^k * trace_k(A) * λ^(n-k).

Cayley-Hamilton theorem?▼

p(A) = 0: substituting matrix A into its characteristic polynomial yields zero matrix.

Degree of characteristic polynomial?▼

Degree equals n for n×n matrix. Leading coefficient is (-1)^n.

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