Characteristic Equation of a Matrix

The characteristic equation of a matrix is an important tool to study a matrix. Using this equation, we can find the eigenvalue with its corresponding eigenvectors to analyse a matrix. In this post, we will learn about the characteristic equation of a matrix.

Definition

Let A be an n×n matrix over a field F. The the equation

det(A-xI_n) = 0

is called the characteristic equation of A where I_n denotes the n×n identity matrix.

Note:

The polynomial det(A-xI_n) is called the characteristic polynomial of A, and it is usually denoted by Ψ_A(x).

If Ψ_A(x) = c₀xⁿ + c₁x^n-1 + …+c_n-1x +c_n is the characteristic polynomial of an n×n matrix A = (a_ij), then we have

Ψ_A(x) = det(A-xI_n)

⇒ c₀xⁿ + c₁x^n-1 + …+c_n-1x +c_n = $\begin{vmatrix} a_{11}-x & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22}-x & \cdots & a_{2n}\\ \cdots & \cdots & \cdots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn}-x \end{vmatrix}$.

From this equality, we obtain that

c0 = (-1)n cr = (-1)n-r × sum of the principal minors of A of order r.

In particular, the coefficients c₁ and c_n are given as follows:

c₁ = (-1)^n-1 (a₁₁ + a₂₂ + … +a_nn) = (-1)^n-1 trace(A).

c_n = det(A).

Example

The characteristic polynomial of I₂ = $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ is given by

det(I₂ – xI₂) = 0

⇒ det $\begin{pmatrix} 1-x & 0 \\ 0 & 1-x \end{pmatrix}$ = 0

⇒ (1-x)² = 0

⇒ 1-2x+x² = 0

So 1-2x+x² = 0 is the characteristic equation of the 2×2 identity matrix I₂.

Also Study:

• Rolle’s Theorem
• nth Derivative

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