Eigenvector of Matrix

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Define Eigenvector

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: 

  1. 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.

  2. AX = λ1 is the formula used to substitute the above values.

  3. The value of eigenvector X is calculated.

  4. 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:

\[X = \begin{bmatrix}x \\ y\end{bmatrix} = k = \begin{bmatrix}1 \\ -1 \end{bmatrix}\]


Characteristics of Eigenvalues

  • Eigenvectors with accurate eigenvalues are linearly independent.

  • Zero eigenvalues are found in single matrices.

  • If A is a square matrix, then A does not have λ = 0 as an eigenvalue.

  • Scalar Multiple of the Matrix: A is a 2 x 2 matrix and the eigenvalue of λ belongs to A.

  • For Matrix Powers: A is a 2 x 2 matrix and the eigenvalue of λ belongs to A where n≥0 is an integer.

  • 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.

  • Inverse Matrix: A is 2 x 2 matrix, the eigenvalue of λ belongs to A  then, λ-1 is an eigenvalue of A - 1.

  • 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:

  1. Left eigenvector.

  2. 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:

  • Used in Physics to study simple oscillations.

  • The eigenvector decomposition is used to solve the first-order linear equations, in matrices ranking and differential calculus.

  • Communication systems.

  • Designing bridges in Civil engineering.

  • Designing a stereo system in a car.

  • Oil companies frequently use eigenvectors and eigenvalues to track oil sightings and mining sites.

FAQ (Frequently Asked Questions)

1. What is the Power Method of Eigenvectors?

Ans: The Power method is used to compute eigenvectors of a matrix (n x n). This method to calculate eigenvectors is usually used in differential calculus.


Let us assume that,

A is a matrix of the order n x n and 

λ = ​λ1, λ2, λ3….λn  are the eigenvalues given where λ1 is the dominant eigenvalue.

We are to select an initial approximate value X0 as a dominant eigenvector of A.

Therefore, X1 = AX0

X2=AX1 which is expanded as AA(X0) = A2X0

Similarly, we have

Xk = AkX0

Therefore, as ‘k’ increases proportionately, a better estimate of the eigenvector is obtained.

2. How are Eigenvectors Used in Linear Mapping?

Ans: Eigenvectors and eigenvalues defined in terms of matrices are used in linear transformations. The eigenvector is a fundamental concept employed in linear mapping strategies as well as various fields in engineering. It is used to measure the level of distortion prompted by the basis of transformation. Eigenvectors explain the orientation and direction of the distortion by using multiple eigenvalues. An approximate visual representation of the Principal component analysis can be drawn from these values. The linear operation used helps to reduce complex problems to simpler ones. This is one of the important statistical procedures utilized in all fields of engineering.


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