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Matrix Minimal Polynomial Calculator

Find the minimal polynomial of a matrix

Enter 2x2 Matrix A

Minimal Polynomial Formula

p_min(x) = product (x - λ_i)^(size of largest Jordan block)

p_min(A) = 0 (annihilates A)

p_min divides p_char (Cayley-Hamilton)

p_min = p_char iff diagonalizable

The minimal polynomial is the monic polynomial of least degree that annihilates the matrix (p(A) = 0). It divides the characteristic polynomial.

⚠ Minimal polynomial is unique for each matrix and contains information about the Jordan structure of the matrix.

What is Minimal Polynomial?

The minimal polynomial of a matrix A is the unique monic polynomial p(x) of least degree such that p(A) = 0. It provides essential information about the matrix's structure and is closely related to its Jordan canonical form.

Monic Polynomial

Leading coefficient is 1. Minimal polynomial is always monic.

Annihilating Polynomial

p(A) = 0. Minimal polynomial is the smallest such polynomial.

Divisor Property

Minimal polynomial divides characteristic polynomial.

Jordan Structure

Exponent of (x-λ) equals size of largest Jordan block.

Teaching Example: A = [[2,1],[0,2]]
1. Characteristic polynomial: p_char(x) = (x-2)^2
2. Check (A-2I) = [[0,1],[0,0]] ≠ 0
3. Check (A-2I)^2 = [[0,0],[0,0]] = 0
4. Minimal polynomial: p_min(x) = (x-2)^2

Applications

Matrix Decomposition Cayley-Hamilton Jordan Form Control Systems Linear Algebra

Frequently Asked Questions

What is minimal polynomial?▼

Monic polynomial of least degree p such that p(A) = 0. Divides characteristic polynomial.

How to find minimal polynomial?▼

Find eigenvalues, determine algebraic/geometric multiplicities, use rational canonical form.

Minimal vs characteristic?▼

Minimal divides characteristic. Equal iff matrix is diagonalizable.

Cayley-Hamilton theorem?▼

Characteristic polynomial annihilates the matrix: p_A(A) = 0.

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