Proof of why multiplication between a matrix and its eigenvector equals the eigenvalue times its eigenvector

Question

Why is the following true?

$$\lambda \vec{x} = P \vec{x} \: \: \: \lambda \:\: \text{is the eigenvalue of P} $$

If the column vectors are the input space, and the row vectors are the output space, why are they related in such a way that the input space is a constant multiple of output space when acting on a eigenvector?

Thanks

Answer 1

You view a $n \times n$ matrix as a linear application $f:\mathbb{R}^n\to \mathbb{R}^n$, so "row" and "columns" have no meaning here.

The rest is a definition