What are Eigenvectors and Eigenvalues in Relation to Matrices?

[Jhenrique]

Given a vector $\vec{r} = x \hat{x} + y \hat{y}$ is possbile to write it as $\vec{r} = r \hat{r}$ being $r = \sqrt{x^2+y^2}$ and $\hat{r} = \cos(\theta) \hat{x} + \sin(\theta) \hat{y}$. Speaking about matrices now, the the eigenvalues are like the modulus of a vector and the eigenvectors are like the unit vectors associated to modulus of a vector?

 

[Mark44]

Given a vector $\vec{r} = x \hat{x} + y \hat{y}$ is possbile to write it as $\vec{r} = r \hat{r}$ being $r = \sqrt{x^2+y^2}$ and $\hat{r} = \cos(\theta) \hat{x} + \sin(\theta) \hat{y}$. Speaking about matrices now, the the eigenvalues are like the modulus of a vector and the eigenvectors are like the unit vectors associated to modulus of a vector?

I don't think these analogies are useful or even correct. The matrices that we're talking about here are square, meaning that they are transformations of some vector space to itself.

The defining relationship between a matrix and its eigenvectors and eigenvalues is this:
Ax = λx, where x is a nonzero vector, and λ is a scalar.

In a sense, the eigenvectors are preferred directions. Any vector with this same direction gets mapped by the transformation to a multiple of itself.

A given transformation from a vector space to itself can have many matrices that represent it, depending on that basis that is used for that space. If an n x n matrix has n distinct eigenvectors, it's possible to write the matrix in terms of this basis of eigenvectors, which results in a diagonal matrix, with the eigenvalues on the main diagonal.