Eigenvector of a 2 × 2 matrix
The Eigenvector of the 2 × 2 matrix can be calculated using the above mention steps. An example of the same is,
Example: Find the eigenvalues and the eigenvector for the matrix A = [Tex]\begin{bmatrix} 1 & 2\\ 5& 4 \end{bmatrix} [/Tex]
Solution:
If eigenvalues are represented using λ and the eigenvector is represented as v = [Tex]\begin{bmatrix} a\\b \end{bmatrix} [/Tex]
Then the eigenvector is calculated by using the equation,
|A- λI| = 0
[Tex]\begin{bmatrix}1 & 2\\ 5& 4\end{bmatrix} -λ\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix} = \begin{bmatrix}0 & 0\\ 0& 0\end{bmatrix} [/Tex]
[Tex]\begin{bmatrix} 1 – λ& 2\\ 5& 4 – λ \end{bmatrix} [/Tex] = 0
(1-λ)(4-λ) – 2.5 = 0
⇒ 4 – λ – 4λ + λ2 – 10 = 0
⇒ λ2 -5λ -6 = 0
⇒ λ2 -6λ + λ – 6 = 0
⇒ λ(λ-6) + 1(λ-6) = 0
⇒ (λ-6)(λ+1) = 0
λ = 6 and λ = -1
Thus, the eigenvalues are 6, and -1. Then the respective eigenvectors are,
For λ = 6
(A-λI)v = 0
⇒ [Tex]\begin{bmatrix}1 – 6& 2\\ 5& 4 – 6\end{bmatrix}.\begin{bmatrix}a\\ b\end{bmatrix} [/Tex] = 0
⇒ [Tex]\begin{bmatrix}-5& 2\\ 5& -2\end{bmatrix}.\begin{bmatrix}a\\ b\end{bmatrix} [/Tex] = 0
⇒ -5a + 2b = 0
⇒ 5a – 2b = 0
Simplifying the above equation we get,
5a=2b
The required eigenvector is,
[Tex]\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}2\\5\end{bmatrix} [/Tex]
For λ = -1
(A-λI)v = 0
⇒ [Tex]\begin{bmatrix}1 – (-1)& 2\\ 5& 4 – (-1)\end{bmatrix}.\begin{bmatrix}a\\ b\end{bmatrix} [/Tex] = 0
⇒ [Tex]\begin{bmatrix}2& 2\\ 5& 5\end{bmatrix}.\begin{bmatrix}a\\ b\end{bmatrix} [/Tex] = 0
⇒ 2a + 2b = 0
⇒ 5a + 5b = 0
simplifying the above equation we get,
a = -b
The required eigenvector is,
[Tex]\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix} 1\\-1\end{bmatrix} [/Tex]
Then the eigenvectors of the given 2 × 2 matrix are [Tex]\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}2\\5\end{bmatrix}, \begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}1\\-1\end{bmatrix} [/Tex]
These are two possible eigen vectors but many of the corresponding multiples of these eigen vectors can also be considered as other possible eigen vectors.
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