Eigenvectors

What is it?

In Linear Algebra, an eigenvector is a vector which is associated with a square matrix, and have multiple interpretations depending on the applied context. In Euclidean Geometry, an eigenvector is such that a matrix-vector multiplication acts like a scalar-vector multiplication. This is expressed as:

$$\mathbf{Av}=\lambda \mathbf{v}$$

These are not equal!

The equality of $\mathbf{Av}=\lambda \mathbf{v}$ expresses that the effect of the matrix $\mathbf{A}$ on the vector $\mathbf{v}$ is the same as the effect of the scalar $\lambda$ on the same vector.

The concept of eigenvectors need Eigenvalues to complement is meaning. The Eigenvalues are expressed as the scalar $\lambda$, while eigenvectors are the vector $\mathbf{v}$.

When working Vector Spaces, eigenvectors will change only by a scalar $\lambda$ after applying a linear transformation to a subspace, and it will not move, but only stretch by the scalar $\lambda$, which is the eigenvalue.