Characteristic Polynomial

In linear algebra, a characteristic polynomial of a square matrix is defined as a polynomial that contains the eigenvalues as roots and is invariant under matrix similarity. The characteristic equation is the equation derived by equating the characteristic polynomial to zero. It is also known as the determinantal equation. Let us look at the definition of characteristic polynomial, formula, and characteristic polynomial of a n×n Matrix, method of finding the Eigenvalues as well as several solved problems in this article.

Table of Contents:

• Definition
  • Characteristic Polynomial of a 2×2 Matrix
  • Characteristic Polynomial of a 3×3 Matrix
• Characteristic Equation
• Roots of Characteristic Polynomial
• FAQs

Characteristic Polynomial Definition

Assume that A is an n×n matrix. Hence, the characteristic polynomial of A is defined as function f(λ) and the characteristic polynomial formula is given by:

f(λ) = det (A – λI_n)

Where I represents the Identity matrix.

The main purpose of finding the characteristic polynomial is to find the Eigenvalues.

Now, let us discuss how to find the characteristic polynomial of 2×2 and 3×3 matrices in the below section:

Characteristic Polynomial of a 2×2 Matrix

As we know, the characteristic polynomial of a matrix A is given by f(λ) = det (A – λI_n).

Now, consider the matrix,

As, the matrix is a 2 × 2 matrix, its identity matrix is,

Now, substitute the values in the characteristic polynomial formula, we get

f(λ) = det (A – λI₂)

On simplifying the above determinant, we get

= (5-λ)(1-λ) – 2(2)

= 5-5λ -λ + λ² – 4

= λ² – 6λ + 1

Hence, the characteristic polynomial of the given 2 × 2 matrix is f(λ) = λ² – 6λ + 1.

Characteristic Polynomial of a 3×3 Matrix

The characteristic polynomial formula for the 3×3 Matrix is given by f(λ) = det (A – λI₃).

Now, let us assume that matrix A is

And, I =

Now, substituting the matrices in the formula, we get

Now, expand the cofactors of the matrix along the third column, we get

f(λ) = 8[(¼) – 0(-λ)] – λ[λ² – 6(½)]

f(λ) = -λ³ + 3λ + 2, which is the characteristic polynomial of the given 3×3 matrix.

Characteristic Equation

If the characteristic polynomial is equated to zero, then the equation obtained is called the characteristic equation.

I.e., f(λ) = 0 (or)

det (A – λI_n) = 0

Where A is an n×n Matrix and I is an identity matrix.

Roots of Characteristic Polynomial

The roots of the characteristic polynomials are the Eigenvalues. The theorem related to this is given below:

Theorem: Assume that A is an n×n matrix, and f(λ) = det (A – λI_n) is a characteristic polynomial, then λ₀ is an Eigenvalue of A, iff f(λ₀) = 0.

Now, let us discuss how to find the Eigenvalues for a given matrix, using the characteristic polynomial.

Consider the same matrix

As we computed above, the characteristic polynomial of the given matrix is f(λ)= λ² – 6λ + 1.

To find the Eigenvalues, we have to solve λ² – 6λ + 1 = 0. ..(1)

By using the quadratic formula, we can find the Eigenvalue (λ).

By comparing (1) with the standard quadratic equation, we get a = 1, b = -6 and c = 1.

Hence,

λ = 3 ± 2√2

Therefore, the Eigenvalues are 3 + 2√2 and 3 – 2√2.