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p

Characteristic Polynomial

p(λ) = det(A − λI). Roots = eigenvalues. For 2×2: λ² − tr(A)λ + det(A) = 0. Coefficients from trace, determinant, and principal minors.

Concept Fundamentals
p(λ)=det(A−λI)
Definition
λ²−tr·λ+det=0
2×2
eigenvalues
Roots
p(A)=0
Cayley-Hamilton

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Eigenvalues = roots of p(λ). tr(A) = sum of roots; det(A) = product. Cayley-Hamilton: every matrix satisfies its char. poly.

Key quantities
p(λ)=det(A−λI)
Definition
Key relation
λ²−tr·λ+det=0
2×2
Key relation
eigenvalues
Roots
Key relation
p(A)=0
Cayley-Hamilton
Key relation

Ready to run the numbers?

Why: Characteristic polynomial yields eigenvalues. Cayley-Hamilton: p(A) = 0. Used in differential equations and control theory.

How: Compute det(A − λI). For 2×2: λ² − (a+d)λ + (ad−bc). For 3×3: expand or use trace/det relations.

Eigenvalues = roots of p(λ).tr(A) = sum of roots; det(A) = product.
Sources:Gilbert Strang, Linear AlgebraWolfram MathWorld

Run the calculator when you are ready.

Compute Characteristic Polynomialp(λ) = det(A − λI)

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What is a Characteristic Polynomial?

The characteristic polynomial of a square matrix A is a polynomial whose roots are the eigenvalues of A. It is defined as p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I), where I is the identity matrix.

This polynomial plays a fundamental role in linear algebra as it helps determine:

  • The eigenvalues of a matrix (as roots of the polynomial)
  • The algebraic multiplicity of each eigenvalue
  • Various matrix properties like trace, determinant, and diagonalizability
  • Solutions to systems of differential equations

The degree of the characteristic polynomial is equal to the dimension of the matrix, and its leading coefficient is always 1.

How to Calculate the Characteristic Polynomial

  1. Form the matrix A - λI by subtracting λ from each diagonal element of A.
    A−λI=[a11−λa12⋯a1na21a22−λ⋯a2n⋮⋮⋱⋮an1an2⋯ann−λ]A - \lambda I = \begin{bmatrix} a_{11} - \lambda & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} - \lambda & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} - \lambda \end{bmatrix}
  2. Calculate the determinant of this matrix, which yields a polynomial in λ.
    p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I)
  3. Expand and simplify the polynomial expression.
  4. Write in standard form: p(λ)=λn+cn−1λn−1+⋯+c1λ+c0p(\lambda) = \lambda^n + c_{n-1}\lambda^{n-1} + \cdots + c_1\lambda + c_0

Special Cases:

For 2×2 matrices:

p(λ)=λ2−tr(A)λ+det⁡(A)p(\lambda) = \lambda^2 - \text{tr}(A)\lambda + \det(A)

Where tr(A) is the trace (sum of diagonal elements) and det(A) is the determinant.

For 3×3 matrices:

p(λ)=λ3−tr(A)λ2+m2λ−det⁡(A)p(\lambda) = \lambda^3 - \text{tr}(A)\lambda^2 + m_2\lambda - \det(A)

Where m₂ is the sum of principal minors of order 2.

Key Properties

Cayley-Hamilton Theorem

Every square matrix satisfies its own characteristic polynomial. If p(λ) is the characteristic polynomial of A, then p(A) = 0.

Coefficient Properties

  • Leading coefficient is always 1
  • Coefficient of λn-1 is -tr(A)
  • Constant term is (-1)ndet(A)

Similar Matrices

Similar matrices have the same characteristic polynomial and therefore the same eigenvalues.

Eigenvalues

The roots of the characteristic polynomial are the eigenvalues of the matrix. Their algebraic multiplicity is the multiplicity of the corresponding root.

Applications

Diagonalization

The characteristic polynomial helps determine if a matrix is diagonalizable and how to find the diagonalization.

Linear Differential Equations

Systems of linear differential equations can be solved using the characteristic polynomial of the coefficient matrix.

Matrix Powers and Functions

Computing high powers or functions of matrices can be simplified using the characteristic polynomial.

Stability Analysis

In dynamical systems and control theory, the stability of systems can be analyzed using the characteristic polynomial.

Historical Context

The concept of the characteristic polynomial emerged in the 19th century during the development of linear algebra and matrix theory. Mathematicians like Augustin-Louis Cauchy and James Joseph Sylvester made significant contributions to its development. The Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic polynomial, was first proved by Arthur Cayley for 3×3 matrices in 1858 and later generalized by William Rowan Hamilton.

Frequently Asked Questions

What is the relationship between eigenvalues and the characteristic polynomial?

The eigenvalues of a matrix are precisely the roots of its characteristic polynomial. The multiplicity of an eigenvalue as a root of the characteristic polynomial is called its algebraic multiplicity.

How is the characteristic polynomial related to the determinant and trace?

For an n×n matrix A, the characteristic polynomial has the form p(λ) = λⁿ + ... + (-1)ⁿdet(A). The coefficient of λⁿ⁻¹ is -tr(A), where tr(A) is the trace (sum of diagonal elements) of the matrix.

Why do similar matrices have the same characteristic polynomial?

If B = P⁻¹AP, then det(B - λI) = det(P⁻¹(A - λI)P) = det(P⁻¹)·det(A - λI)·det(P) = det(A - λI), since det(P⁻¹)·det(P) = 1. This shows that similar matrices have identical characteristic polynomials.

What is the Cayley-Hamilton theorem?

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial. If p(λ) = det(A - λI) is the characteristic polynomial of matrix A, then p(A) = 0 (the zero matrix).

How are characteristic polynomials used in practice?

Characteristic polynomials are used in various applications, including solving systems of differential equations, analyzing dynamical systems, determining matrix functions, checking diagonalizability, and computing matrix powers efficiently.

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

p

deg(p) = n for n×n matrix

📐

p(A) = 0 (Cayley-Hamilton)

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