What are Eigenvalues and Eigenvectors?
If A is a square matrix of order n × n then we can easily find the eigenvector of the square matrix by following the method discussed below,
We know that the eigenvector is given using the equation Av = λv, for the identity matrix of order same as the order of A i.e. n × n we use the following equation,
(A-λI)v = 0
Solving the above equation we get various values of λ as λ1, λ2, …, λn these values are called the eigenvalues and we get individual eigenvectors related to each eigenvalue.
Simplifying the above equation we get v which is a column matrix of order n × 1 and v is written as,
[Tex]v = \begin{bmatrix} v_{1}\\ v_{2}\\ v_{3}\\ .\\ .\\ v_{n}\\ \end{bmatrix} [/Tex]
Eigenvalues
Eigenvalues and Eigenvectors are the scalar and vector quantities associated with Matrix used for linear transformation. The vector that does not change even after applying transformations is called the Eigenvector and the scalar value attached to Eigenvectors is called Eigenvalues. Eigenvectors are the vectors that are associated with a set of linear equations. For a matrix, eigenvectors are also called characteristic vectors, and we can find the eigenvector of only square matrices. Eigenvectors are very useful in solving various problems of matrices and differential equations.
In this article, we will learn about eigenvalues, eigenvectors for matrices, and others with examples.
Table of Content
- What are Eigenvalues?
- What are Eigenvectors?
- Eigenvector Equation
- What are Eigenvalues and Eigenvectors?
- How to Find an Eigenvector?
- Types of Eigenvector
- Right Eigenvector
- Left Eigenvector
- Eigenvectors of a Square Matrix
- Eigenvector of a 2 × 2 matrix
- Eigenvector of a 3 × 3 Matrix
- Eigenspace
- Appliactions of Eigen Values
- Diagonalize Matrix Using Eigenvalues and Eigenvectors
- Solved Examples on Eigenvectors
- FAQs on Eigenvectors