Eigenvector of a matrix is also known as latent vector, proper vector or characteristic vector. These are defined in the reference of a square matrix.Matrix is an important branch that is studied under linear algebra. Matrix is a rectangular array of numbers or other elements of the same kind. It generally represents a system of linear equations.
A very useful concept related to matrices is EigenVectors. It is vector that is associated with a set of linear equations. These are also useful in solving differential equations and many other applications related to them. Let us go ahead and understand eigenvector, how to find eigenvalue of a 2×2 matrix, its technique and various other concepts related to it.
The method of determining eigenvector of a matrix is given below:
If A be an n×n matrix and λ be the eigenvalues associated with it. Then, eigenvector v can be defined by the following relation:
Av = λv
If I be the identity matrix of same order as A, then
Eigenvector associated with matrix A can be determined using above method.
Here, v is known as eigenvector belonging to each eigenvalue and is written as:
The equation corresponding to each eigenvalue of a matrix is given by:
AX = λX
It is formally known as eigenvector equation.
In place of λ, we one by one put each eigenvalue and get the eigenvector equation which enables us to solve for eigen vector belonging to each eigenvalue.
For example: Suppose that there are two eigenvalues λ1 = 0 and λ2 = 1 of any 2 × 2 matrix.
AX = λ1X,
AX = O …..(1)
AX = λ2X,
A = 1,
(A–I)X = O …. (2)
Equations (1) and (2) are eigen vector equations for given matrix.
Where, I = Identity matrix of same order as A
O = zero matrix of same order as AX = Eigen vector which is equal to [xy]
(as A is of order 2)
How to Find Eigenvector
In order to find eigenvectors of a matrix, one needs to follow the following given steps:
Step 1: Determine the eigenvalues of given matrix A using the equation det (A – λI) = 0, where I is equivalent order identity matrix as A. Denote each eigenvalue of λ1, λ2, λ3, …
Step 2: Substitute the value of λ1 in equation AX = λ1X or (A – λ1I) X = O.
Step 3: Calculate the value of eigenvector X which is associated with eigenvalue λ1.
Step 4: Repeat steps 3 and 4 for other eigenvalues λ2, λ3, … as well.
There is a little difference between eigenvector and generalized eigenvector. It is defined in the following way:
A generalized eigenvector associated with an eigenvalue λ of an n×n matrix is denoted by a nonzero vector X and is defined as:
(A−λI)k = 0
Where, k is some positive integer.
For k = 1 ⇒(A−λI) = 0
Therefore, if k = 1, then eigenvector of matrix A is its generalized eigenvector.
We know that a vector quantity possesses magnitude as well as direction. So, an eigenvector has some magnitude in a particular direction. Orthogonality is a concept of two eigenvectors of a matrix being perpendicular to each other. We can say that when two eigenvectors make a right angle between each other, these are said to be orthogonal eigenvectors.
A symmetric matrix (in which aij=aji) does necessarily have orthogonal eigenvectors.
There are basically two types of eigenvectors:
1) Left Eigenvector
2) Right Eigenvector
Left eigenvector is a type of eigenvector that is represented in the form of a row vector which satisfies the following condition:
Where, A is given matrix of order n and λ be one of its eigenvalue. XL is denoted by row vector [x1x2…xn]
In the same way as left eigenvector, right eigenvector is denoted by XR. It is defined as an eigonvector that is written in the form of a column vector, satisfying the condition given below:
In which, A denotes an n×n square matrix and represents it eigenvalue.
Power Method for Eigenvectors
Power method is an important method for computing eigenvectors of a matrix. It is an iterative method used in numerical analysis. Power method works in the following way –
Let us assume that A be a matrix of order n x n and λ1,λ2,…,λn be its eigenvalues, such that λ1 be the dominant eigenvalue. We are to select an initial approximate value x0 for a dominant eigenvector of A.
The system associated with the eigenvalue λ = 0 is (A−0I)⎝⎜⎛xyz⎠⎟⎞=⎝⎜⎛1000101311360⎠⎟⎞⎝⎜⎛xyz⎠⎟⎞=⎝⎜⎛000⎠⎟⎞
There are different applications of eigenvectors in real life. Some of the important ones are illustrated below:
1) In mathematics, eigenvector decomposition is widely used in order to solve linear equations of first order, in ranking matrices, in differential calculus etc.
2) Eigenvectors are used in physics in simple mode of oscillation.
3) This concept is widely used in quantum mechanics.
4) Eigenvectors are widely applicable in almost all the branches of engineering.
5) Eigenvalues can be used to calculate the theoretical limit of the transmission of information through a medium of communication like a telephone line or through the air. It is done by finding the eigenvalues and eigenvectors of the channel that is used for communication and then water-filling on the eigenvalues.
6) Eigenvalue is the natural frequency that is the smallest magnitude of a system that builds the bridge. The stability of the construction can be monitored by using this.
7) The concept of eigenvalues is used in the construction of a car stereo system. It helps in the reproduction of the vibration of the car due to the music.
8) It is used in decoupling three-phase systems along the symmetrical
9) It helps in the reduction of a linear operation to separate simpler problems.
10) Many oil companies generally use eigenvalue analysis to survey land for oil.
11) It is used to explain natural occurrences.
12) The Linear mapping eigenvalues measures the distortion induced by the transformation and eigenvectors explains about the orientation of the distortion. It gives a rough picture of the Principal Component Analysis = A statistical procedure.
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