Eigen Values & Eigen Vectors

# Eigen Values & Eigen Vectors | Engineering Mathematics - Civil Engineering (CE) PDF Download

## Definition

Eigenvalues are the special set of scalars associated with the system of linear equations. It is mostly used in matrix equations. ‘Eigen’ is a German word that means ‘proper’ or ‘characteristic’. Therefore, the term eigenvalue can be termed as characteristic value, characteristic root, proper values or latent roots as well. In simple words, the eigenvalue is a scalar that is used to transform the eigenvector. The basic equation is
Ax = λx
The number or scalar value “λ” is an eigenvalue of A.

In Mathematics, an eigenvector corresponds to the real non zero eigenvalues which point in the direction stretched by the transformation whereas eigenvalue is considered as a factor by which it is stretched. In case, if the eigenvalue is negative, the direction of the transformation is negative.

For every real matrix,  there is an eigenvalue. Sometimes it might be complex. The existence of the eigenvalue for the complex matrices is equal to the fundamental theorem of algebra.

### What are EigenVectors?

Eigenvectors are the vectors (non-zero) that do not change the direction when any linear transformation is applied. It changes by only a scalar factor. In a brief, we can say, if A is a linear transformation from a vector space V and x is a vector in V, which is not a zero vector, then v is an eigenvector of A if A(X) is a scalar multiple of x.

An Eigenspace of vector x consists of a set of all eigenvectors with the equivalent eigenvalue collectively with the zero vector. Though, the zero vector is not an eigenvector.

Let us say A is an “n × n” matrix and λ is an eigenvalue of matrix A, then x, a non-zero vector, is called as eigenvector if it satisfies the given below expression;
Ax = λx
x is an eigenvector of A corresponding to eigenvalue, λ.

Note:

There could be infinitely many Eigenvectors, corresponding to one eigenvalue.

For distinct eigenvalues, the eigenvectors are linearly dependent.

### Eigenvalues of a Square Matrix

Suppose, An×n is a square matrix, then [A- λI] is called an Eigen or characteristic matrix, which is an indefinite or undefined scalar. Where determinant of Eigen matrix can be written as, |A- λI| and |A- λI| = 0 is the Eigen equation or characteristics equation, where “I” is the identity matrix. The roots of an Eigen matrix are called Eigen roots.

Eigenvalues of a triangular matrix and diagonal matrix are equivalent to the elements on the principal diagonals. But eigenvalues of the scalar matrix are the scalar only.

### Properties of Eigenvalues

• Eigenvectors with Distinct Eigenvalues are Linearly Independent
• Singular Matrices have Zero Eigenvalues
• If A is a square matrix, then λ = 0 is not an eigenvalue of A
• For a scalar multiple of a matrix: If A is a square matrix and λ is an eigenvalue of A. Then, aλ is an eigenvalue of aA.
• For Matrix powers: If A is square matrix and λ is an eigenvalue of A and n ≥ 0 is an integer, then λn is an eigenvalue of An.
• For polynomials of matrix: If A is a square matrix, λ is an eigenvalue of A and  p(x) is a polynomial in variable x, then p(λ) is the eigenvalue of matrix p(A).
• Inverse Matrix: If A is a square matrix, λ is an eigenvalue of A, then λ-1 is an eigenvalue of A-1
• Transpose matrix: If A is a square matrix, λ is an eigenvalue of A, then λ is an eigenvalue of At

### EigenValue Example

In this shear mapping, the blue arrow changes direction, whereas the pink arrow does not. Here, the pink arrow is an eigenvector because it does not change direction. Also, the length of this arrow is not changed; its eigenvalue is 1.

### Eigenvalues of 2 x 2 Matrix

Let us have a look at the example given below to learn how to find the eigenvalues of a 2 x 2 matrix.
Find the eigenvalues of the 2 x 2 matrix

Given,

Using the characteristic equation,

Let

be the 2 x 2 identity matrix.

|A – λI| = 0

-λ(4 – λ) – (-2)(0) = 0
-4λ + λ2 = 0
λ(λ – 4) = 0
λ = 0 or λ – 4 = 0
Thus, λ = 0 or λ = 4
Hence, the two eigenvalues of the given matrix are λ = 0 and λ = 4.
Go through the following problem to find the Eigenvalue of 3 x 3 matrix.

### 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.

### 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 with 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 =

Solution: We have already solved for the eigenvalues and the eigenvectors of the A =

The eigenvalues of the A are λ = 0, λ = 0, and λ = -8

The eigenvectors of A are

Thus, D = , X =

We can easily find the inverse of X as, X-1

The document Eigen Values & Eigen Vectors | Engineering Mathematics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mathematics.
All you need of Civil Engineering (CE) at this link: Civil Engineering (CE)

## Engineering Mathematics

65 videos|120 docs|94 tests

## FAQs on Eigen Values & Eigen Vectors - Engineering Mathematics - Civil Engineering (CE)

 1. What are eigenvalues and eigenvectors?
Eigenvalues and eigenvectors are concepts in linear algebra. Eigenvalues are scalar values that represent the scaling factor of eigenvectors in a linear transformation. Eigenvectors, on the other hand, are non-zero vectors that remain in the same direction after the linear transformation.
 2. How do eigenvalues and eigenvectors relate to matrices?
Eigenvalues and eigenvectors are closely related to matrices. Given a square matrix, the eigenvectors are the vectors that do not change direction under the linear transformation represented by the matrix. The eigenvalues are the corresponding scalar values that describe the scaling factor of these eigenvectors.
 3. How can eigenvalues and eigenvectors be calculated?
To calculate eigenvalues and eigenvectors, one needs to solve the characteristic equation of the matrix. The characteristic equation is obtained by subtracting the identity matrix multiplied by a scalar λ from the original matrix, and then taking the determinant of the resulting matrix. Solving the characteristic equation yields the eigenvalues, and substituting these eigenvalues back into the original equation helps find the corresponding eigenvectors.
 4. What are the applications of eigenvalues and eigenvectors?
Eigenvalues and eigenvectors have numerous applications in various fields. In physics, they are used to describe the behavior of oscillating systems, such as vibrations of a string or a pendulum. In computer science, they are utilized in data analysis and machine learning algorithms, such as principal component analysis (PCA) and spectral clustering. They are also employed in image processing, network analysis, and quantum mechanics, among other areas.
 5. Can a matrix have multiple eigenvalues and eigenvectors?
Yes, a matrix can have multiple eigenvalues and eigenvectors. The number of eigenvalues is equal to the dimension of the matrix, and the eigenvectors corresponding to distinct eigenvalues are linearly independent. However, multiple eigenvectors can exist for a single eigenvalue, forming an eigenspace. This occurs when a matrix has repeated eigenvalues or when the eigenvectors span a larger subspace.

## Engineering Mathematics

65 videos|120 docs|94 tests

### Up next

 Explore Courses for Civil Engineering (CE) exam

### Top Courses for Civil Engineering (CE)

Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;