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.
What are Eigenvalues?
Eigenvalues are the scalar values associated with the eigenvectors in linear transformation. The word ‘Eigen’ is of German Origin which means ‘characteristic’. Hence, these are the characteristic value that indicates the factor by which eigenvectors are stretched in their direction. It doesn’t involve the change in the direction of the vector except when the eigenvalue is negative. When the eigenvalue is negative the direction is just reversed. The equation for eigenvalue is given by
Av = λv
where A is the matrix
v is associated eigenvector
λ is scalar eigenvalue
What are Eigenvectors?
Eigenvectors for square matrices are defined as non-zero vector values which when multiplied by the square matrices give the scaler multiple of the vector, i.e. we define an eigenvector for matrix A to be “v” if it specifies the condition, Av = λv
The scaler multiple λ in the above case is called the eigenvalue of the square matrix. We always have to find the eigenvalues of the square matrix first before finding the eigenvectors of the matrix.
For any square matrix, A of order n × n the eigenvector is the column matrix of order n × 1. If we find the eigenvector of the matrix A by, Av = λv, “v” in this is called the right eigenvector of the matrix A and is always multiplied to the right-hand side as matrix multiplication is not commutative in nature. In general, when we find the eigenvector it is always the right eigenvector.
We can also find the left eigenvector of the square matrix A by using the relation, vA = vλ
Here, v is the left eigenvector and is always multiplied to the left-hand side. If matrix A is of order n × n then v is a column matrix of order 1 × n.
Eigenvector Equation
The Eigenvector equation is the equation that is used to find the eigenvector of any square matrix. The eigenvector equation is,
Av = λv
where,
A is the given square matrix
v is the eigenvector of matrix A
λ is any scaler multiple
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]
How to Find an Eigenvector?
The eigenvector of the following square matrix can be easily calculated using the steps below,
Step 1: Find the eigenvalues of the matrix A, using the equation det |(A – λI| =0, where “I” is the identity matrix of order similar to matrix A
Step 2: The value obtained in Step 2 are named as, λ1, λ2, λ3….
Step 3: Find the eigenvector (X) associated with the eigenvalue λ1 using the equation, (A – λ1I) X = 0
Step 4: Repeat step 3 to find the eigenvector associated with other remaining eigenvalues λ2, λ3….
Following these steps gives the eigenvector related to the given square matrix.
Types of Eigenvector
The eigenvectors calculated for the square matrix are of two types which are,
- Right Eigenvector
- Left Eigenvector
Right Eigenvector
The eigenvector which is multiplied by the given square matrix from the right-hand side is called the right eigenvector. It is calculated by using the following equation,
AVR = λVR
where
A is given square matrix of order n×n
λ is one of the eigenvalues
VR is the column vector matrix
The value of VR is,
[Tex]V_{R} = \begin{bmatrix} v_{1}\\ v_{2}\\ v_{3}\\ .\\ .\\ v_{n}\\ \end{bmatrix}
[/Tex]
Left Eigenvector
The eigenvector which is multiplied by the given square matrix from the left-hand side is called the left eigenvector. It is calculated by using the following equation,
VLA = VLλ
where
A is given square matrix of order n×n
λ is one of the eigenvalues
VL is the row vector matrix
The value of VL is,
VL = [v1, v2, v3,…, vn]
Eigenvectors of a Square Matrix
We can easily find the eigenvector of square matrices of order n×n. Now, let’s find the following square matrices:
- Eigenvectors of a 2 × 2 matrix
- Eigenvectors of a 3 × 3 matrix.
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.
Eigenvector of a 3 × 3 Matrix
The Eigenvector of the 3 × 3 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} 2 & 2 & 2 \\ 2 & 2 & 2\\2 & 2 & 2 \end{bmatrix}
[/Tex]
Solution:
If eigenvalues are represented using λ and the eigenvector is represented as v = [Tex]\begin{bmatrix} a\\b\\c \end{bmatrix}
[/Tex]
Then the eigenvector is calculated by using the equation,
|A- λI| = 0
[Tex]\begin{bmatrix}2 & 2 & 2\\ 2 & 2 & 2\\ 2 & 2 & 2\end{bmatrix} -λ\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\0 & 0 & 1\end{bmatrix} = \begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix}
[/Tex]
[Tex]\begin{bmatrix} 2 – λ & 2 & 2 \\ 2 & 2 – λ & 2 \\ 2 & 2 & 2- λ\end{bmatrix}
[/Tex] = 0
Simplifying the above determinant we get
(2-λ)(λ2) + 2λ2 + 2λ2 = 0
(-λ3) + 6λ2 = 0
λ2(6 – λ) = 0
λ = 0, λ = 6
For λ = 0
(A – λI) v = 0
[Tex]\begin{bmatrix}2 – 0& 2& 2\\ 2& 2 – 0&2\\2 & 2 & 2-0\end{bmatrix}.\begin{bmatrix}a\\ b\\c\end{bmatrix}
[/Tex] = 0
[Tex]\begin{bmatrix}2& 2& 2\\ 2& 2 &2\\2 & 2 & 2\end{bmatrix}.\begin{bmatrix}a\\ b\\c\end{bmatrix}
[/Tex] = 0
Simplifying the above equation we get
2a + 2b + 2c = 0
2(a+b+c) = 0
a+b+c = 0
Let b = k1 and c = k2
a + k1 + k2 = 0
a = -(k1 + k2)
Thus, the eigenvector is,
[Tex]\begin{bmatrix}a\\ b\\c\end{bmatrix} = \begin{bmatrix}-(k_{1}+k_{2})\\ k_{1}\\k_{2}\end{bmatrix}
[/Tex]
taking k1 = 1 and k2 = 0
the eigenvector is,
[Tex]\begin{bmatrix}a\\ b\\c\end{bmatrix} = \begin{bmatrix}-1\\ 1\\0\end{bmatrix}
[/Tex]
taking k1 = 0 and k2= 1
the eigenvector is,
[Tex]\begin{bmatrix}a\\ b\\c\end{bmatrix} = \begin{bmatrix}-1\\ 0\\1\end{bmatrix}
[/Tex]
For λ = 6
(A – λI) v = 0
[Tex]\begin{bmatrix}2 – 6& 2& 2\\ 2& 2 -6&2\\2 & 2 & 2-6\end{bmatrix}.\begin{bmatrix}a\\ b\\c\end{bmatrix}
[/Tex] = 0
[Tex]\begin{bmatrix}-4& 2& 2\\ 2& -4 &2\\2 & 2 & -4\end{bmatrix}.\begin{bmatrix}a\\ b\\c\end{bmatrix}
[/Tex] = 0
Simplifying the above equation we get,
-4a +2b +2c = 0
2 (-2a + b + c) = 0
-2a = – (b + c)
2a = b + c
Let b = k1 and c = k2 , and taking k1 = k2 = 1,
we get,
[Tex]\begin{bmatrix}a\\ b\\c\end{bmatrix} = \begin{bmatrix}1\\ 1\\1\end{bmatrix}
[/Tex]
Thus, the eigenvector is,
[Tex]\begin{bmatrix}a\\ b\\c\end{bmatrix} = \begin{bmatrix}1\\ 1\\1\end{bmatrix}
[/Tex]
Eigenspace
We define the eigenspace of a matrix as the set of all the eigenvectors of the matrix. All the vectors in the eigenspace are linearly independent of each other.
To find the Eigenspace of the matrix we have to follow the following steps
Step 1: Find all the eigenvalues of the given square matrix.
Step 2: For each eigenvalue find the corresponding eigenvector.
Step 3: Take the set of all the eigenvectors (say A). The resultant set so formed is called the Eigenspace of the following vector.
From the above example of given 3 × 3 matrix A, the eigenspace so formed is{ [Tex]\begin{bmatrix}0\\ 0\\0\end{bmatrix},\begin{bmatrix}-1\\ 0\\-1\end{bmatrix},\begin{bmatrix}-1\\ -1\\0\end{bmatrix}
[/Tex] }
Appliactions of Eigen Values
Linear Algebra:
Diagonalization: Eigenvalues are used to diagonalize matrices, simplifying computations and solving linear systems more efficiently.
Matrix Exponentiation: Eigenvalues play a crucial role in computing the exponentiation of a matrix.
Quantum Mechanics:
Schrödinger Equation: Eigenvalues of the Hamiltonian operator correspond to the energy levels of quantum systems, providing information about possible states.
Vibrations and Structural Analysis:
Mechanical Vibrations: Eigenvalues represent the natural frequencies of vibrational systems. In structural analysis, they help understand the stability and behavior of structures.
Statistics:
Covariance Matrix: In multivariate statistics, eigenvalues are used in the analysis of covariance matrices, providing information about the spread and orientation of data.
Computer Graphics:
Principal Component Analysis (PCA): Eigenvalues are used in PCA to find the principal components of a dataset, reducing dimensionality while retaining essential information.
Control Systems:
System Stability: Eigenvalues of the system matrix are critical in determining the stability of a control system. Stability analysis helps ensure that the system response is bounded.
Diagonalize Matrix Using Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors are used to find diagonal matrices. A diagonal matrix is a matrix which can be written as,
A = XDX-1
where,
D is the matrix which is formed by replacing the 1’s in the identity matrix by eigenvalues.
X is the matrix formed by eigenvectors.
We can understand the concept of a diagonal matrix by taking the following example.
Example: Diagonalize the matrix A = [Tex]\begin{bmatrix} 2 & 2 & 2 \\ 2 & 2 & 2\\2 & 2 & 2 \end{bmatrix}
[/Tex]
Solution:
We have already solved for the eigenvalues and the eigenvectors of the A = [Tex]\begin{bmatrix} 2 & 2 & 2 \\ 2 & 2 & 2\\2 & 2 & 2 \end{bmatrix}
[/Tex]
The eigenvalues of the A are λ = 0, λ = 0, and λ = -8
The eigenvectors of A are [Tex]\begin{bmatrix}0\\ 0\\0\end{bmatrix},\begin{bmatrix}-1\\ 0\\-1\end{bmatrix},\begin{bmatrix}-1\\ -1\\0\end{bmatrix}
[/Tex]
Thus,
D = [Tex]\begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & -8\end{bmatrix}
[/Tex]
X = [Tex]\begin{bmatrix}0 & -1 & -1\\0 & 0 & -1\\0 & -1 & 0\end{bmatrix}
[/Tex]
We can easily find the inverse of X as,
X-1 = [Tex]\begin{bmatrix}0 & -1 & -1\\0 & 0 & -1\\0 & -1 & 0\end{bmatrix}
[/Tex]
Read More,
Solved Examples on Eigenvectors
Example 1: Find the eigenvectors of the matrix A = \begin{bmatrix}1 & 1 & 0\\0 & 1 & 1\\0 & 0 & 1\end{bmatrix}
Solution:
The eigen values of the matrix is found using,
|A – λI| = 0
[Tex]\begin{bmatrix}1-λ & 1 & 0\\0 & 1-λ & 1\\0 & 0 & 1-λ\end{bmatrix}
[/Tex] = 0
(1 – λ)3 = 0
Thus, the eigen values are,
λ = 1, 1, 1
As the all the eigenvalues are equal we have three identical eigenvectors. We will find the eigenvectors for λ = 1, using (A – λI)v = O
[Tex]\begin{bmatrix}1-1 & 1 & 0\\0 & 1-1 & 1\\0 & 0 & 1-1\end{bmatrix}.\begin{bmatrix}a\\ b\\c\end{bmatrix} = \begin{bmatrix}0\\ 0\\0\end{bmatrix}
[/Tex]
[Tex]\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix}.\begin{bmatrix}a\\ b\\c\end{bmatrix} = \begin{bmatrix}0\\ 0\\0\end{bmatrix}
[/Tex]
solving the above equation we get,
Then the eigenvector is,
[Tex]\begin{bmatrix}a\\ b\\c\end{bmatrix}= \begin{bmatrix}k\\ 0\\0\end{bmatrix} = k\begin{bmatrix}1\\ 0\\0\end{bmatrix}
[/Tex]
Example 2: Find the eigenvectors of the matrix A = [Tex]\begin{bmatrix}5 & 0\\0 & 5 \end{bmatrix}
[/Tex]
Solution:
The eigen values of the matrix is found using,
|A – λI| = 0
[Tex]\begin{bmatrix}5-λ & 0\\0 & 5-λ \end{bmatrix}
[/Tex] = 0
(5 – λ)2 = 0
Thus, the eigen values are,
λ = 5, 5
As the all the eigenvalues are equal we have three identical eigenvectors. We will find the eigenvectors for λ = 1, using
(A – λI)v = O
[Tex]\begin{bmatrix}5-5 & 0 \\ 0 & 5-5\end{bmatrix}.\begin{bmatrix}a\\ b\end{bmatrix} = \begin{bmatrix}0\\ 0\end{bmatrix}
[/Tex]
Simplying the above we get,
Then the eigenvector is,
[Tex]\begin{bmatrix}a\\ b\end{bmatrix}= \begin{bmatrix}1\\ 0\end{bmatrix} , \begin{bmatrix}0\\ 1\end{bmatrix}
[/Tex]
FAQs on Eigenvectors
Q1: What are Eigenvectors?
Answer:
We define the eigenvector of any matrix as the vector which on multiplying with the matrix results in the scaler multiple of the matrix.
Q2: How to find Eigenvectors?
Answer:
Eigenvector of any matrix A is denoted by v. Eigenvector of the matrix is calculated by first finding the eigenvalue of the matrix.
- Eigenvalue of the matrix is found using the formula, |A-λI| = 0 where λ gives the eigenvalues.
- After finding eigenvalue we found eigenvector by the formula, Av = λv, where v gives the eigenvector.
Q3: What is the difference between Eigenvalue and Eigenvector?
Answer:
For any square matrix A the eigenvalues are represented by λ and it is calculated by the formula, |A – λI| = 0. After finding the eigenvalue we find the eigenvector by, Av = λv.
Q4: What is the Diagonalizable Matrix?
Answer:
Any matrix that can be expressed as the product of the three matrices as XDX-1 is a diagonalizable matrix here D is called the diagonal matrix.
Q5: Are Eigenvalues and Eigenvectors same?
Answer:
No, Eigenvalues and Eigenvectors are not same. Eigenvalues are the scaler which is used to find eigenvectors whereas eigenvectors are the vectors that are used to find matrix vector transformations.
Q6: Can Eigenvector be a Zero Vector?
Answer:
We can have eigenvalues be zero but the eigenvector can never be a zero vector.
Q7: What is Eigenvectors Formula?
Answer:
The eigenvector of any matrix is calculated using the formula,
Av = λv
where,
λ is the eigenvalue
v is the eigenvector
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