What are eigenvalues and eigenvectors?

by | Oct 3, 2023 | Featured, Linear Algebra

Eigen Values And Eigen Vectors

What are eigenvalues and eigenvectors?

Let’s say A is a square matrix. Now, if we multiply A with a vector u, we will get another vector v.

A \vec{u}= \vec{v}

The direction of the vectors u and v will be different in most cases. In other words, if we multiply a matrix with a vector, the direction and magnitude of the vector change. But, if u is an eigenvector of the square matrix A, then the vector v and the vector u will point to the same direction. In other words, the square matrix will scale the vector u.

A \vec{u}= \lambda \vec{u}

The vector u is called an eigenvector of the square matrix A and λ is called an eigenvalue.

So, we can say from the above equation,

(A- \lambda I) \vec{u}=0

In other words, if u is not a null vector, then we can get the eigenvalues of a square matrix A by solving the following equation:

det(A- \lambda I)=0

If we solve for λ, we get the eigenvalues. And for a particular eigenvalue λ, when we solve the following equation, we get the eigenvector corresponding to the eigenvalue.

(A- \lambda I) \vec{u}=0

How to calculate the eigenvalues and the eigenvectors of a square matrix?

Now, let’s look at an example. Let’s say A is the following square matrix…

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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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