Eigenvector of Matrix

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:

A_X = λ_X

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 [ x_1, x_2, x_3,…. X_n]

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 [ x_1, x_2, x_3, …. X_n].

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.