What is the characteristic polynomial of a matrix?

by | Oct 3, 2023 | Featured, Linear Algebra

Characteristic Polynomial

Let A be the following 3×3 square matrix. And I be the identity matrix of dimension 3×3.

A=\begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 0 \\ 0 & 1 & 0 \end{bmatrix}

Now, if we subtract λI from A, we will get the following:

(A- \lambda I)=\begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 0 \\ 0 & 1 & 0 \end{bmatrix}- \lambda \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ =\begin{bmatrix} {1- \lambda } & 0 & 2 \\ 2 & {1- \lambda } & 0 \\ 0 & 1 & {- \lambda } \end{bmatrix}

Now, if we take the determinant of (A – λI), we will get a polynomial f(λ).

f( \lambda )=det(A- \lambda I) \\ =(1- \lambda) (- \lambda (1- \lambda) - 0)-0+2 \times (2-0) \\ =- \lambda (1- \lambda)^2+4

This polynomial is called the characteristic polynomial of the matrix A.

Please note that if we calculate the roots of the equation f(λ) = 0, we get the eigenvalues of A.

 

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Amrita Mitra

Author

Ms. Amrita Mitra is an author, who has authored the books “Cryptography And Public Key Infrastructure“, “Web Application Vulnerabilities And Prevention“, “A Guide To Cyber Security” and “Phishing: Detection, Analysis And Prevention“. She is also the founder of Asigosec Technologies, the company that owns The Security Buddy.

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