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 = λ1XAX = O …..(1)
AX = λ2XA = 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.