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The eigenvector is referred to as a vector that is associated with a set of linear equations. A latent vector, appropriate vector, or characteristic vector is the eigenvector of a matrix. These are defined using a square matrix as a reference. Eigenvectors are also useful in the solution of differential equations and a variety of other applications. Matrix's Eigenvector is also known as a latent vector. It has a square matrix and is associated with linear algebraic equations. In this article, we will be explaining more about the eigenvector of matrix and discuss some important questions.
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Keywords: Eigenvector, Matrix, Determinants, Matrix, Ncert, Eigenvector Applications, Eigenvector Equation, Eigen Value of Matrix
Also read: Isosceles Triangle Theorems
Eigenvector Definition
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The eigenvector of a square matrix is defined as a non-vector that is equivalent to a scalar multiple of a given vector when the given matrix is multiplied. If v is a non-zero vector and A is a n x n square matrix, then the product of matrix A and vector v is defined as the product of a scalar quantity λ with the supplied vector, as follows:
Av =λv
Where
v = Eigenvector and λ are the scalar quantity that is called as eigenvalue associated with given matrix A
Eigenvector Formula
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Each eigenvalue of a matrix has an equation that goes like this:
AX = λ X
The eigenvector equation is the official name for it.
Substitute each eigenvalue in place of λ, to obtain the eigenvector equation, which allows us to find the eigenvector associated with each eigenvalue.
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Eigenvector Method
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The following is the procedure for calculating the eigenvector of a matrix:
If A is a (nxn) matrix, and λ be the eigenvalues of A. The following relation may then be used to define eigenvector v:
AV = λV
If "I" is the same-order identity matrix as A, then
(A–λI)v = 0
The aforementioned approach may be used to get the eigenvector associated with matrix A.
The eigenvector of each eigenvalue is denoted by "v," which is written as:
v = [ v1 v2 v3 ........ vn ]
How to Determine Eigenvector?
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Follow the steps below to determine the eigenvectors of a matrix:
- Use the equation det ((A –λI) =0 to get the eigenvalues of the given matrix A, where "I" is the equivalent order identity matrix to A and eigenvalues can be denoted as λ1, λ2, and λ3a and so on.
- Substitute the values in the equation AX = λ1 or (A – λ1 I) X = 0 for the values in the equation AX = 1.
- Calculate the eigenvector X value, which is linked to the eigenvalue.
- For the remaining eigenvalues, repeat the procedures to obtain the eigenvector.
Let's use the example of a n x n matrix named 'A' and the associated eigenvalues λ to get the eigenvector.
The set of eigenvalues known as 'v' (v = v1, v2, v3.. vn) may thus be represented as Av =λv
If " I " the specified identity matrix (n x n) is comparable to matrix 'A,' then: (A- λI)v = 0.
As a result, the above equation may be used to find the eigenvector for matrix A.
Types of Eigenvectors
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There are two types of eigenvectors:
- Left Eigenvector
- Right Eigenvector
Left Eigenvector
The left eigenvector is expressed as a row vector that meets the following requirements:
AXL=λXL
Where:-
A is a given n-Order matrix, and λ is one of its eigenvalues.
A matrix's row vector is XL. [Example: x1 x2 x3.... Xn]
XLA = λLXL
Right Eigenvector
The right eigenvector is written as a column vector that meets the following requirements:
AXR = λXR
Where:-
A is a given n-dimensional matrix, and λ is one of its eigenvalues.
XR is a matrix column vector.
AXR = λRXR
Eigenvector Applications
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The following are some of the most common uses of eigenvectors:-
- In Physics, eigenvectors are utilised in the basic mode of oscillation.
- In mathematics, eigenvector decomposition is commonly used to solve first-order linear equations, ranking matrices, and differential calculus, among other things.
- Quantum mechanics makes extensive use of this idea.
- Communication systems.
- Civil engineering includes the design of bridges.
- Designing a car stereo system.
- Eigenvectors and eigenvalues are widely used by oil firms to monitor oil sightings and mining locations.
- Facial Recognition
- Stability analysis - physics of a rotating body
- Google uses it to rank pages for your search result.
Matrix Eigenvalues
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In linear algebra, eigenvalues are commonly connected with eigenvectors. In the interpretation of linear transformations, both of these phrases are utilized. Eigenvalues are a specific collection of scalar values associated with linear equations, most commonly in matrix equations, as we all know.
We must first find eigenvectors before we can define eigenvalues. When a vector is multiplied by A, almost all of them change direction. Some uncommon vectors say x is pointing in the same direction as Ax. These are referred to as "eigenvectors." When you multiply an eigenvector by A, you get the vector Ax, which is the number of times the original x occurred. The fundamental equation is as follows:
Ax = λx.
The number λ is an eigenvalue of matrix A in this case.
Things to Remember
- Linear independence exists between eigenvectors with exact eigenvalues.
- In single matrices, zero eigenvalues can be discovered.
- If A is a square matrix, then A does not have an eigenvalue as λ = 0.
- Scalar Multiple of the Matrix: The eigenvalue of λ belonging to A & A is a 2 x 2 matrix.
- For Matrix Powers: The eigenvalue of λ belongs to A where n≥0 is an integer and A is a 2 x 2 matrix.
- For a Polynomial of the Matrix: The eigenvalue of λ belongs to A, where A is 2 x 2 matrix, and p(x) is the polynomial belonging to variable x.
- Inverse Matrix: If A is a 2 x 2 matrix, and the eigenvalue of λ belongs to A then, λ-1 is an eigenvalue of A - 1.
- Transpose Matrix: If A is a 2 x 2 matrix, the eigenvalue of λ belongs to A then λ is an eigenvalue of At.
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Sample Questions
Ques: Find the eigenvector of the given matrix: A=[14−4−7] (4 marks)
Ans: Given:
A=[14−4−7]
(1- λ)(-7- λ)- 4(-4) = 0
( λ+3)2 = 0
Therefore, λ =-3, -3
Use the eigenvector equation
AX = λX
Substitute λ value in the equation:
AX = -3X
We know that,
(A- λI) X = 0
([14−4−7]+[3003])[xy]=[00]
4x +4y =0
Or,
x+y =0
Assume that x =k
So, it becomes
k +y =0
y= -k
Therefore, the eigenvector is
X=[xy]=k[1−1]
Now, let’s understand how we can find the eigenvalue of the matrix along with a solved example here.
Ques: Determine the eigenvalues and eigenvectors of the identity matrix I without first calculating its characteristic equation. (4 marks)
Ans: The equation A x = λ x characterizes the eigenvalues and associated eigenvectors of any matrix A.
If A = I, this equation becomes x = λ x. Since x ≠ 0, this equation implies λ = 1; then, from x = 1 x, every (nonzero) vector is an eigenvector of I.
Remember the definition: x is an eigenvector of a matrix A if A x is a scalar multiple of x and x ≠ 0.
Since multiplication by I leaves x unchanged, every (nonzero) vector must be an eigenvector of I, and the only possible scalar multiple—eigenvalue—is 1.
Ques: Let A be a square matrix. How do the eigenvalues and associated eigenvectors of A 2 compare with those of A? Assuming that A is invertible, how do the eigenvalues and associated eigenvectors of A −1 compare with those of A? (4 marks)
Ans: Let λ be an eigenvalue of the matrix A, and let x be a corresponding eigenvector. Then A x = λ x, and it follows from this equation that
\(A^2x = A(Ax) = A(\lambda x)= \lambda (Ax) = \lambda (\lambda x)= \lambda ^2x\)Therefore, λ 2 is an eigenvalue of A 2, and x is the corresponding eigenvector. Now, if A is invertible, then A has no zero eigenvalues, and the following calculations are justified:
\(Ax = \lambda x A^{-1} (Ax) = A^{-1} (\lambda x) x = \lambda (A^{-1} x) \lambda ^{-1}x = A^{-1} x\)so λ −1 is an eigenvalue of A −1 with corresponding eigenvector x.
Ques: Determine the eigenvectors of the matrix: 
(5 marks)
Ans: The eigenvalues of this matrix were found to be λ = −1 and λ = −2. Therefore, there are nonzero vectors x such that A x = x (the eigenvectors corresponding to the eigenvalue λ = −1), and there are nonzero vectors x such that A x = −2 x (the eigenvectors corresponding to the eigenvalue λ = −2). The eigenvectors corresponding to the eigenvalue λ = −1 are the solutions of the equation A x = −x:
This is equivalent to the pair of equations
x1 – 2x2 = -x1
3x1 – 4x2 = -x1
which simplifies to
2x1 – 2x2 = 0
3x1 – 3x2 = 0
[Note that these equations are not independent. If they were independent, then only ( x 1, x 2) T = (0, 0) T would satisfy them; this would signal that an error was made in the determination of the eigenvalues. If the eigenvalues are calculated correctly, then there must be nonzero solutions to each system A x = λ x.]
The equations above are satisfied by all vectors x = ( x 1, x 2) T such that x 2 = x 1. Any such vector has the form ( x 1, x 2) T. and is, therefore, a multiple of the vector (1, 1) T. Consequently, the eigenvectors of A corresponding to the eigenvalue λ = −1 are precisely the vectors
where t is any nonzero scalar.
The eigenvectors corresponding to the eigenvalue λ = −2 are the solutions of the equation A x = −2 x:
This is equivalent to the “pair” of equations 3x1 – 2x2 = 0
3x1 – 2x2 = 03x1 – 2x2 = 0
Again, note that these equations are not independent. They are satisfied by any vector x = ( x 1, x 2) T that is a multiple of the vector (2, 3) T; that is, the eigenvectors of A corresponding to the eigenvalue λ = −2 are the vectors
where t is any nonzero scalar.
Ques: Let , 
Find its eigenvalues and eigenvectors. (4 marks)
Ans: At first we find eigenvalues of A by solving the equation:
\(det (\lambda I - A) = 0\)this gives,
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