Eigen Vector
An Eigen Vector is defined as any vector $ \vec v $ which can satisfy:
$ A\vec v = \lambda \vec v $
for $ \lambda $ is a scalar called an Eigen Value and $ A $ is a square matrix.
Purpose
Eigen Vectors are valuable tools in both linear algebra and solving systems of differential equations. In linear algebra Eigen Vectors can be used to find the diagonalized form of a matrix.
Finding an Eigen Vector
The easiest way to find Eigen Vectors is to first determine the Eigen Values ($ \lambda_1,\lambda_2,\cdots,\lambda_k $). Then, simply substitute each Eigen Value and solve for the kernel of the new matrix:
$ (\lambda_jI_n-A)\vec v = \vec 0 $
for each Eigen Value where $ I_n $ is the $ n^{th} $ identity matrix. Since in finding the Eigen Values required making $ det(\lambda_jI_n-A)=0 $, that means that the solution to the above equation cannot be the the zero space. Therefore, there will be vectors which span the space, and the space is called an Eigen Space. Any vector contained in the Eigen Space is an Eigen Vector. The dimension of the Eigen Space determines the number of linearly independent Eigen Vectors can be found for a corresponding Eigen Value. The dimension of the Eigen Space is also called the geometric multiplicity of its corresponding Eigen Value. There is a relationship between the algebraic multiplicity of an Eigen Value and its geometric multiplicity.
algebraic multiplicity >= geometric multiplicity >= 1 for all Eigen Values.
Diagonalizable
A matrix or linear transformation is said to be diagonalizable if a similar matrix can be found that is a diagonal matrix. This means that the transformations of the vectors of the basis of the domain are scalar multiples of the corresponding base vector. In other words, for linear transformation $ T $ with matrix $ A $ and a proposed diagonal basis $ \vec v_1,\vec v_2,\cdots,\vec v_n $.
$ T(\vec v_j)=A\vec v_j=c\vec v_j $
This means that $ \vec v_j $ is an Eigen Vector of A and $ c $ is an Eigen Value.
Therefore, if there is as many Eigen Vectors as there are Eigen Values, the matrix is diagonalizable. In other words, the algebraic multiplicity of Eigen Values must be the same as the geometric multiplicity for every Eigen Value.
Finally, by the known methods of changing the basis of a matrix, it can be seen that the resulting diagonal matrix will be:
$ D=\big[\lambda_1e_1,\lambda_2e_2,\cdots,\lambda_ne_n\big] $
where $ e_j $ is the $ j^{th} $ standard basis.
