Characteristic Polynomial and Minimal Polynomial of a Matrix

Introduction

In linear algebra, the characteristic polynomial and minimal polynomial of a matrix are fundamental tools that encapsulate essential properties of the matrix. The characteristic polynomial, denoted $ p_A(\lambda) $, is defined as the determinant of the matrix $ A - \lambda I $, where $ I $ is the identity matrix and $ \lambda $ is a scalar variable. This polynomial provides critical information about the eigenvalues of the matrix, as its roots correspond to the eigenvalues of $ A $. The minimal polynomial, denoted $ m_A(\lambda) $, is the monic (leading coefficient 1) polynomial of least degree such that $ m_A(A) = 0 $. This polynomial captures the structure of the matrix's Jordan canonical form and is closely tied to the eigenvalues and their multiplicities. Together, these polynomials form the backbone of matrix theory, enabling the analysis of matrix properties such as diagonalizability, similarity, and solvability of systems of linear equations.

Definition and Properties of the Characteristic Polynomial

The characteristic polynomial of a square matrix $ A $ of size $ n \times n $ is given by:
$$ p_A(\lambda) = \det(A - \lambda I) $$
This polynomial is of degree $ n $ and is uniquely determined by the matrix $ A $. The roots of $ p_A(\lambda) $, denoted $ \lambda_1, \lambda_2, \ldots, \lambda_n $, are the eigenvalues of $ A $. Additionally, the coefficients of $ p_A(\lambda) $ are related to the trace and determinant of $ A $, as follows:

  • The coefficient of $ \lambda^{n-1} $ is equal to $ -\text{tr}(A) $, the trace of $ A $.
  • The constant term $ (-1)^n \det(A) $ is the determinant of $ A $.

The characteristic polynomial is also closely tied to the concept of invariant subspaces. For instance, if $ \lambda $ is an eigenvalue of $ A $, then $ \lambda $ is a root of $ p_A(\lambda) $, and the algebraic multiplicity of $ \lambda $ in $ p_A(\lambda) $ corresponds to the geometric multiplicity of $ \lambda $ in the matrix $ A $.

Definition and Properties of the Minimal Polynomial

The minimal polynomial of a matrix $ A $ is the monic polynomial of least degree such that $ m_A(A) = 0 $. This polynomial is uniquely determined by the structure of $ A $ and is closely related to the Jordan canonical form of $ A $. For a given matrix $ A $, the minimal polynomial satisfies the following properties:

  1. Monic and Irreducible: The minimal polynomial is monic and irreducible over the field $ \mathbb{F} $, where $ \mathbb{F} $ is the field of coefficients.
  2. Roots and Eigenvalues: The roots of $ m_A(\lambda) $ are the eigenvalues of $ A $, and the multiplicity of each root in $ m_A(\lambda) $ corresponds to the size of the largest Jordan block associated with that eigenvalue.
  3. Degree and Matrix Size: The degree of the minimal polynomial is at most $ n $, the size of the matrix $ A $, and is equal to the size of the largest Jordan block in the Jordan canonical form of $ A $.

For example, consider a matrix $ A $ with a single Jordan block of size $ k $ corresponding to an eigenvalue $ \lambda $. The minimal polynomial of $ A $ is $ (x - \lambda)^k $, and its degree is $ k $. If $ A $ is diagonalizable, the minimal polynomial is the product of $ (x - \lambda) $ for each distinct eigenvalue $ \lambda $, and its degree is equal to the number of distinct eigenvalues.

Relationship Between Characteristic and Minimal Polynomials

The minimal polynomial of a matrix $ A $ divides its characteristic polynomial. This relationship is crucial in understanding the structure of $ A $. Specifically, if $ m_A(\lambda) $ divides $ p_A(\lambda) $, then $ m_A(\lambda) $ is the monic polynomial of least degree that annihilates $ A $. This divisibility property ensures that the minimal polynomial captures the essential structure of the matrix, particularly its eigenvalues and their multiplicities.

The degree of the minimal polynomial is directly related to the size of the matrix and the nature of its Jordan form. For instance, if $ A $ is diagonalizable, the minimal polynomial is the product of $ (x - \lambda) $ for each distinct eigenvalue $ \lambda $, and its degree equals the number of distinct eigenvalues. Conversely, if $ A $ has a Jordan block of size $ k $ corresponding to an eigenvalue $ \lambda $, the minimal polynomial is $ (x - \lambda)^k $, and its degree is $ k $.

Moreover, the minimal polynomial and the characteristic polynomial share the same set of eigenvalues, but the minimal polynomial may have higher multiplicities for certain eigenvalues. This distinction is critical in determining the matrix's Jordan form and its diagonalizability. For example, if the minimal polynomial contains a factor $ (x - \lambda)^k $, then $ A $ has at least $ k $ Jordan blocks corresponding to $ \lambda $, and the characteristic polynomial will have a root $ \lambda $ with multiplicity at least $ k $.

Applications in Matrix Theory

The characteristic and minimal polynomials find wide applications in matrix theory, including the study of diagonalization, similarity transformations, and the solution of systems of linear differential equations.

  1. Diagonalization and Jordan Form: The minimal polynomial of a diagonalizable matrix is the product of $ (x - \lambda) $ for each distinct eigenvalue $ \lambda $, and its degree equals the number of distinct eigenvalues. If the minimal polynomial is $ \prod_{i=1}^k (x - \lambda_i) $, then the matrix is diagonalizable, and the characteristic polynomial is the product of $ (x - \lambda_i)^{m_i} $, where $ m_i $ is the algebraic multiplicity of $ \lambda_i $.

  2. Cayley-Hamilton Theorem: This theorem states