The eigenvector definition is based on the concept of matrices. An eigenvector is described as a non-vector wherein the matrix given is multiplied and equated to the scalar multiple of the said vector. This is calculated precisely for a square matrix. Assuming ‘A’ is a square matrix of n x n and the non-zero vector is taken is ‘v’, then the product of vector ‘v’ and matrix ‘A’ would be described as the product of the stated vector and scalar quantity λ.
Av = λv
Wherein,
v = Eigenvector
λ = the scalar quantity is known as the eigenvalue associated with matrix A.
To Find Eigenvectors of Matrix
To find the Eigenvector of a matrix, the following steps are employed:
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The eigenvalues for matrix A are found by using the formula, det (A – λI) = 0. Here, ‘I’ is defined as the equivalent of the order of the matrix identity ‘A’. Further, eigenvalues can be denoted as λ1, λ2, and λ3.
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AX = λ1 is the formula used to substitute the above values.
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The value of eigenvector X is calculated.
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The above steps are repeated to obtain the remaining eigenvectors.
Finding Eigenvectors
To find the eigenvector, let’s take the example of an n x n matrix named ‘A’ and ‘λ’ be the associated eigenvalues provided.
Then the set of eigenvalues termed collectively as ‘v’ (v = v1, v2, v3.. vn) can be described as,
Av = λv
If ‘I’ is the provided identity matrix (n x n) similar to matrix ‘A’, then:
(A- λI)v = 0
Therefore, the eigenvector related to matrix A can be calculated using the above equation.
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Example of Using an Eigenvalue
Find the eigenvalues using the given matrix.
[A = begin{bmatrix}-5 &2 \-7 &4 end{bmatrix}]
Ans:
If [A = begin{bmatrix}-5 &2 \-7 &4 end{bmatrix}], then in terms of the formula det(A-λI)=0,
Then, [A – lambda I = begin{bmatrix}-5 &2 \-7 &4 end{bmatrix}]
We know that, A-λI = 0
|A – λI| = 0
[begin{bmatrix}lambda + 5 &-2 \7 &lambda-4 end{bmatrix} = 0]
[lambda^{2} + lambda – 6 = 0]
Therefore, λ1 = 2 and λ2 = -3
What Does The Eigenvector of The Matrix Mean?
The Eigenvector of Matrix is referred to as a latent vector. It is associated with linear algebraic equations and has a square matrix.
To calculate the eigenvector of a given matrix, the formula is described as follows:
AX = λX
Here, λ is substituted with given eigenvalues to obtain the eigenvector for a set of matrices.
Example of Calculating The Eigenvector of a Matrix
To find the eigenvector for the below matrix,
[A = begin{bmatrix}1 &4 \-4 &-7 end{bmatrix}]
If [A = begin{bmatrix}1 &4 \-4 &-7 end{bmatrix}], then in terms of the formula det(A-λI) = 0,
Then, A-λI = [A = begin{bmatrix}1 &4 \-4 &-7 end{bmatrix}]
We know that, A-λI = 0
|A-λI| = 0
[A = begin{bmatrix}lambda – 1 &-4 \4 &lambda + 7 end{bmatrix} = 0]
[(lambda + 3)^{2} = 0]
AX = λ X
AX = -3X
We know that, |A-λI|X = 0
[(begin{bmatrix}1 &4 \-4 &-7 end{bmatrix} + begin{bmatrix}3 &0 \0 &3 end{bmatrix})begin{bmatrix}x \ y end{bmatrix} = 0]
Therefore, x + y = 0 since 4x + 4y = 0
If x = k, then
x – k = 0
x = -k.
Therefore, the eigenvector is as follows:
[X = begin{bmatrix}x \ yend{bmatrix} = k = begin{bmatrix}1 \ -1 end{bmatrix}]
Characteristics of Eigenvalues
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Eigenvectors with accurate eigenvalues are linearly independent.
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Zero eigenvalues are found in single matrices.
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If A is a square matrix, then A does not have λ = 0 as an eigenvalue.
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Scalar Multiple of the Matrix: A is a 2 x 2 matrix and the eigenvalue of λ belongs to A.
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For Matrix Powers: A is a 2 x 2 matrix and the eigenvalue of λ belongs to A where n≥0 is an integer.
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For a Polynomial of the Matrix: A is 2 x 2 matrix, the eigenvalue of λ belongs to A, and p(x) is the polynomial belonging to variable x.
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Inverse Matrix: A is 2 x 2 matrix, the eigenvalue of λ belongs to A then, λ-1 is an eigenvalue of A – 1.
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Transpose Matrix: A is a 2 x 2 matrix, the eigenvalue of λ belongs to A then λ is an eigenvalue of At.
Orthogonality of an Eigenvector
This is defined as the perpendicular nature of two eigenvector matrices. These can be of two types:
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Left eigenvector.
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Right eigenvector.
a) Left Eigenvector
This is a row vector that follows the condition stated below.
AXL = λXL.
Here, A represents the stated matrix of order n and λ is an eigenvalue.
XL represents a row vector matrix [ x1, x2, x3,…. Xn]
b) Right Eigenvector
This is a column vector that follows the condition stated below.
AXR = λXR
Here, A represents the stated matrix of order n.
λ is an eigenvalue.
XL represents a column vector matrix [ x1, x2, x3, …. Xn].
Applications of an Eigenvector
Eigenvectors and eigenvalues are used in the following ways:
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Used in Physics to study simple oscillations.
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The eigenvector decomposition is used to solve the first-order linear equations, in matrices ranking and differential calculus.
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Communication systems.
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Designing bridges in Civil engineering.
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Designing a stereo system in a car.
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Oil companies frequently use eigenvectors and eigenvalues to track oil sightings and mining sites.