Difference between Eigenvalue and Eigenvector

Let A be a square matrix of order ‘n-by-n.’ A scalar k is called a eigenvalue of A, if there exist a non-zero vector v satisfying Av = kv, then v is called eigenvector corresponding to eigenvalue k. In this article, we will briefly discuss eigenvalues and eigenvectors and how to find them.

In this article, we will discuss one of the most essential topics of Linear Algebra, i.e., Eigenvalues and Eigenvectors. But before the start of the article, we will discuss Linear Transformation, which we will use in the article.

Linear Transformation

Linear Transformation (T) is a function between two vector spaces ( let V and W over a field F) satisfying:

1. T (v 1 + v 2) = T(v 1) + T(v 2), for v 1 and v 2 belonging to V
2. T(kv) = kT(v), where k is a constant ( or scalar)

They are mainly used for graphic designing, gaming, and statistic, not limited to scaling the vectors; they can rotate, flip, and shear the vectors.

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Table of Content

• Eigen Value and Eigen Vector
• Properties of Eigenvalue and Eigenvector
• Proof of Formula for determining Eigenvalues
• Eigenspace
• Examples
  • Eigenvalues of 2 x 2 matrix
  • Eigenvalues of 3 x 3 matrix
• Eigenvector and Eigenspace for 3 x 3 matrix
• Application of Eigenvectors

Eigen Value and Eigen Vector

Let T be a Linear Transformation from a vector space V to itself over a field F, then any non-zero vector v in V is known as the eigenvector of T; if T(v) is a scalar multiple of v, that can be written as:

T(v) = kv……………….(i)

Where k is a scalar in F, also known as Eigen Value or Characteristic Value.

Confused!!
Let’s make it simple, and understand it in terms of Matrix ( as every linear transformation can be written as a matrix)

Let A be a square matrix of order ‘n-by-n.’ A scalar k is called an Eigen Value of A if there exists a non-zero vector (column) v such that

Av = kv…………………(ii)

Any vector satisfying the above relation is called an eigenvector corresponding to the eigenvalue k.

Note:

Eigen is a german word that means “Proper” or “Characteristic”; hence, we can call the eigenvalue and eigenvector characteristic value.

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Properties of Eigenvalue and Eigenvector

• Eigenvalues and eigenvectors are for square matrices only.
• By definition, an eigenvector can’t be zero, but an eigenvalue may or may not be zero.
  • ‘0’ is an eigenvalue of any matrix ‘A’ if and only if A is not invertible.
• There can be infinitely many eigenvectors corresponding to an eigenvalue.
• An ‘n-by-n’ matrix has at most n-eigenvalues.
• Eigenvectors with distinct eigenvalues are always linearly independent.
• Eigenvalue of matrix A = Eigenvalue of transpose of A
• Sum of eigenvalue of matrix A = sum of diagonal element of matrix A
• Product of eigenvalue of matrix A = determinant of matrix A
• If A an B are two matrices of same order then,
  • Eigenvalue of AB = Eigenvalue of BA

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How to calculate the Eigenvalue

Now to find the formula for determining the eigenvalue, we will use the equation (ii):

Av = kv

Subtracting kv from both the side, we get:

(here I is an Identity matrix)

Now, equation (iii) can be equal to zero in only two possible ways:

(i) v = 0

(ii) (A – kI) = 0

But we know that v must be a non-zero vector, i.e., v = 0 is not possible.

So, (A – kI) should always be zero for v to be an eigenvector.

Now, we will solve (A – kI) using the determinant of the matrix:

Therefore,

det(A – kI) = 0 or |A-kI| = 0…….(iv)

The above equation is also known as the Characteristic Equation.

We will now take some examples to understand better how to find eigenvalue. But before that, we will check out another important topic related to eigenvalue and eigenvector, i.e., Eigenspaces.

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Eigenspace

An eigenspace is a collection of eigenvectors corresponding to eigenvalues. Eigenspace can be extracted after plugging the eigenvalue value in the equation (A-kI) and then normalizing the matrix element.

Eigenspace provides all the possible eigenvector corresponding to the eigenvalue.

Eigenspaces have practical uses in real life:

• Vibration Analysis – Eigenspace describes the shape of the vibration model of an object for each eigenvalue or natural frequency 
• Geology – It can be used to summarize the 3-D space orientation

Examples

Eigenvalue of 2 x 2 Matrix

Example 1: Find the eigenvalue of the below 2 x 2 matrix.

Now, we will use the equation (iv) to find the eigenvalue:

Let k is an eigenvalue of A, then

Hence, there are two different eigenvalues of A (k1 = -7 and k2 = 6).

Eigenvalue of 3 x 3 Matrix

Example 2: Find the eigenvalue of the below 3 x 3 matrix.

Let k be an eigenvalue of the above matrix, now using (iv), we will get:

Hence, the above matrix has three eigenvalues: two are complex numbers, and one is a real number.

Eigenvector and Eigenspace for 3 x 3 matrix

Now, we will find the eigenvector and eigenspace of a 3 x 3 matrix, so let’s consider a 3 x 3 matrix:

Hence, the eigen values for the matrix C is -5, 3, and 6.

Now, we will find eigenvector corresponding to all the the eigenvalues.

To find the eigenvector, we will use the (A – kI)v = 0, where v = (x, y, z) is an eigenvector and k is an eigenvalue. So, we will substitute the eigenvalue in the above equation and solve the linear equation to find the value of (x, y, z).

Eigenvector corresponding to eigenvalue -5

Now, solving the above we get x = 2, y = 1, and z = -1.

Eigenvector corresponding to eigenvalue 3

Now, solving the above we get x = 2, y = -3, and z = –1.

Eigenvalue corresponding to eigenvalue 6

Solving the above, we get x = 1, y = 6, and z = 16.

From above we have three different eigenvectors corresponding to three different eigenvalues.

Hence, the eigenspace corresponding to the eigenvalues are:

k₁ = E_k1 = vector (2, 1, -1), k₂ = E_k2 = vector (2, -3, -2), and k₃ = E_k3 = vector (1, 6, 16).

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Application of Eigenvector

Concept of eigenvalue and eigenvector are used in several fields such as machine learning, quantum computing, construction design, electrical and mechanical engineering.

1. Machine Learning: Concept of eigenvalue and eigenvector are used in feature extraction technique such as Principal Component Analysis (PCA).

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2. Electrical and Mechanical Engineering: In electrical engineering it is use to decouple three-phase system whereas in mechanical engineering eigenvalue and eigenvector enable us to decompose into smaller and more manageable tasks.

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3. Oil companies frequently use the concept to explore land for oil.

Conclusion

In this article, we have briefly discussed eigenvalue and eigenvector, their properties and how to find eigenvalue and eigenvector.

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