Eigenvalues and Eigenvectors and their Examples

Eigenvalues and eigenvectors are some of the most significant concepts of the structure theory of square matrices. As a result, eigenvalues and eigenvectors are likely to play a key role in the real-life applications of linear algebra.

What are Eigenvalues and Eigenvectors?
How to find Eigenvalues and Eigenvectors?
Properties of Eigenvalues
Solved Examples
Frequently Asked Questions

What are Eigenvalues and Eigenvectors?

Eigenvalues are a distinctive group of scalars that make up an arrangement of linear equations. It is usually involved in matrix equations. The German word "Eigen" means "proper" or "characteristic." Thus, the term "eigenvalue" can also be used to refer to a suitable value, a latent root, a characteristic value, or a characteristic root. In simple terms, the eigenvalue is a scalar that is applied to convert the eigenvector.

Eigenvectors are non-zero vectors whose orientation does not change when a linear change is made. It only alters by a single scalar factor.

An eigenvector is equivalent to real non-zero eigenvalues pointing in a particular direction of the transformation's additional direction, whereas an eigenvalue is believed to be the variable that determines how far it is extended.

Let us suppose there is a formula matrix called A.

There exists a nonzero vector v in Rⁿ such that Av = λv, for some scalar . This vector is called the Eigenvector of A.
There exists a scalar such that the equation Av = λv has a non-trivial solution. The scalar is called the Eigenvalue of A.

If Av = λv for, v ≠ 0 and we say that

is the eigenvalue for v, and

v is the eigenvector for .

How to find Eigenvalues and Eigenvectors?

To find the eigenvalues of a matrix A, we use the formula:

formula

Where I indicates an identity matrix.

Solving for formula , we will get the values of the eigenvalues. A 2 × 2 matrix has 2 eigenvalues. A 3 × 3 matrix has 3 eigenvalues.

Now, we put the value of in the equation:

formula

Solving the matrix, we will get the eigenvectors for the corresponding eigenvalues.

Properties of Eigenvalues

There are certain properties of Eigenvalues, such as the following:

If a matrix A has an eigenvalue , then any matrix A^k where k is a positive integer will have an eigenvalue .
A matrix's trace (the total of each element on its major diagonal) equals the sum of its eigenvalues.
Singular matrices have zero eigenvalues.
A square matrix's determinant, when multiplicities are taken into account, is equivalent to the product of its eigenvalues. As a result, the determinant is zero if and only if a minimum of one eigenvalue is zero.
There is linear independence between eigenvectors that correspond to different eigenvalues.

Solved Examples

formula

Which is not a scalar multiple of w. Hence, w it is not an eigenvector of A. formula formula formula

Frequently Asked Questions

Q1. Can eigenvalues be zero?
Answer: Yes, Eigenvalues can be zero.

Q2. Can a single matrix have eigenvalues?
Answer: Every single matrix has a 0 eigenvalue.

Q3. What is the difference between an eigenvector and an eigenvalue?
Answer: Any matrix ‘A’ can be represented as Av = λv where v is called the eigenvector of A for λ. Similarly, the scalar quantity λ is called the eigenvalue of A for the vector v.