Characteristic Polynomial: Formula & Examples

In the realm of linear algebra, the characteristic polynomial stands as a fundamental concept that bridges matrices and their eigenvalues. 

• These polynomials will have eigenvalues as roots and are invariant under matrix similarity. 
• A characteristic equation is a type of equation which is determined by equating the characteristic polynomial to zero.
• This equation is also known as the deterministic equation.
• It not only serves as a tool for theoretical exploration but also has practical applications in areas such as differential equations, control theory, and more.
• The characteristic polynomial is used to calculate closed forms for the solutions of linear recurrences.
• With this article, we can easily learn about the implications of the characteristic polynomial for 2×2 and 3×3 matrices, along with the characteristic equation and the significance of its roots.

Key Terms: Characteristic Polynomial, Characteristic Equation, Matrices, Polynomial, Deterministic Equation, Eigenvalues, Roots of Characteristic Polynomial

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Characteristic Polynomial Definition

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The characteristic polynomial is a powerful tool in linear algebra that provides insights into the properties of matrices. 

• By understanding how to compute and interpret the characteristic polynomial and its roots, we can gain deeper insights into the behaviour of linear transformations and systems.
• Let’s consider (A) to be a square matrix of order (n). The characteristic polynomial of (A), denoted by f(λ), is expressed as:

\(f(λ) = det (A – λI_n)\)

\(f(λ)=(−1)^nλ^n+(−1)^{n−1}tr(A)λn−1+···+det(A).\)

• Here, (I) symbolizes the Identity matrix of the same order.
• The primary objective of computing the characteristic polynomial is to determine the eigenvalues of the matrix.
• When we talk about the spectral graph theory then the characteristic polynomial of a graph is represented by the characteristic polynomial of the adjacency matrix.

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Characteristic Polynomial of a 2 x 2 Matrix

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For a 2×2 matrix (A), the characteristic polynomial can be derived by subtracting (λ) times the identity matrix from (A) and calculating the determinant of the result. The characteristic polynomial of a 2 x 2 matrix is given as:

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Characteristic Polynomial of a 3×3 Matrix

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For a 3×3 matrix, the process is similar but involves a more complex calculation due to the larger size of the matrix. The characteristic polynomial of a 3 x 3 matrix is given as:

\(−λ^3+tr(A)λ^2−\frac{1}{2}(tr(A)^2−tr(A^2))λ+det(A)\)

• Where tr is the trace of matrix A

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Characteristic Equation

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The characteristic equation is obtained by setting the characteristic polynomial equal to zero:

\(det(A−λI) = 0 \;or \;f(λ) = 0\)

• Solving this equation yields the eigenvalues of the matrix (A).

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Roots of Characteristic Polynomial

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The roots of the characteristic polynomial are the eigenvalues of the matrix. These values are fundamental as they often reveal much about the nature of the matrix, including its invertibility, diagonalizability, and stability properties.

Theorem

Consider n x n matrix A whose characteristic polynomial is given as \(f(λ) = det (A – λI_n)\); then the eigenvalue of A is \(λ_0\) where \(f(λ_0) = 0\).

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Things to Remember

• The characteristic polynomial is invariant under matrix similarity.
• The degree of these polynomials is equal to the size of the matrix.
• For a 2 × 2 and 3 x 3 matrix, the characteristic polynomial is quadratic and cubic respectively.
• The trace and determinant of a matrix are integral in finding these types of polynomials.
• The characteristic equation is used to find the eigenvalues of a matrix.
• Understanding the characteristic polynomial is crucial for applications in various scientific and engineering disciplines.

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