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Introduction to eigenvector example: An Eigenvector is defined as a non-zero vector in which we can’t change its direction by a given linear transformation. Linear transformation can be denoted as T. Linear transformation can be given as follows, T(v)= `lambdav` .The other name of Eigen vector is characteristic vector . Eigen vector produces the scalar multiplication of the original vector. Eigenvector has a wide range of applications in all over the fields.In this article we are going to see some examples for eigen vector. Computation of eigen vector example: • A linear transformation T: Rn that tends to Rn given by an n x n matrix B. The Eigen value l and the eigenvector v of T can be defined by Bv = lv. • Unvaryingly, v is a vector that has in null space (B- lI). The number l and the vector v are also called the Eigen value and the eigenvector of B. • The following proposes may helps to find the Eigen values. • l is an Eigen value of matrix B. • Bv = lv where v should not be equal to zero. • (B-lI)x = 0.that has a non trivial solution x=v. • B-lI is non invertible. • Determination of B-lI = 0. • The characteristic polynomial of a given square matrix B is det(B-lI). • Thus the Eigen values and the eigenvectors can be work out as follows. Step 1: Get the Eigen values l1 and l2 by calculating the characteristics equation. Step 2: For each Eigen value l solve the homogeneous system B-lI = 0. and get the eigenvectors with li as the Eigen value. Example Problems for Eigenvector: Eigen vector example 1: If that B is a matrix and that inverse matrix of B is B^-1 and if that y is an eigenvector for matrix B with the Eigen value is `[[2,1],[4,4]]` ? 0. Prove that y is an eigenvector for inverse matrix B^-1 with the Eigen value `[[2,1],[4,4]]`^-1 (inverse of matrix `[[2,1],[4,4]]`). Solution: Let us assume B.y = c, therefore: y = B^-1 c Where B is a matrix When a matrix B and a nonzero vector y satisfy: B.y = `[[2,1],[4,4]]` y (for some scalar matrix `[[2,1],[4,4]]`), and therefore y = B^-1 c, Then we get the value of y as follows, y= `[[2,1],[4,4]]`-1.y, therefore: `[[2,1],[4,4]]`^-1.y = B^-1.y Eigen vector example 2: Consider the following 2x2 matrix `[[2,-1],[0,3]]` . Find all the eigenvectors that are related to the Eigen value `lambda=3` Solution: In the above shown example we have verified that in actuality `lambda=3` is an Eigen value of the given matrix. Let Y0 be an eigenvector that are related to the Eigen value `lambda=3`. Set Y0= `[[x0,],[yo,]]`. Then we have the following equations (2-3)x0 + -y0 = 0. 0 + (3-3)y0 = 0. which reduces to the only equation -x0-y0 = 0. This yields y = -x. Therefore, we have Y0= `[[x0,],[yo,]]` = `[[x0,],[-xo,]]` Y0=x0 `[[1,],[-1,]]` Remain that we are all having all of the eigenvectors that are related to the Eigen value `lambda=3` . Understand more on about How to Multiply Square Roots and its Illustrations. Between, if you have issue on these subjects Is Pi a Rational Number?, Please discuss your feedback.
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