An eigen vector is a non-zero vector that, when multiplied by a square matrix, results in a scalar multiple of itself. It represents the direction in which the matrix has a simple behavior, much like the x and y axes in a graph.
Eigen vectors are important in linear algebra because they help us understand the behavior of a matrix. They represent the directions in which the matrix has a simple behavior, and can be used to decompose a matrix into simpler parts for easier analysis.
To find eigen vectors, you first need to find the eigenvalues of a matrix. This can be done by solving the characteristic equation det(A-λI)=0. Once you have the eigenvalues, you can plug them back into the equation (A-λI)x=0 to find the eigen vectors.
Yes, a matrix can have multiple eigen vectors. In fact, most matrices have multiple eigen vectors. However, eigen vectors must be linearly independent, meaning they cannot be scaled versions of each other.
Eigen vectors are used in the process of diagonalization, which involves finding a diagonal matrix that is similar to a given matrix. The columns of the diagonal matrix are the eigen vectors of the original matrix. This process is useful for simplifying calculations and solving systems of equations.