Define eigenvalue and eigenvector. Distinguish between regression and interpolation. [2.5+2.5]

--- (a) Eigenvalue and Eigenvector Let $A$ be an $n \times n$ square matrix. A scalar $\lambda$ is called an eigenvalue of $A$ if there exists a non-zero vector $\mathbf{x}$ such that: $$A\mathbf{x} = \lambda\mathbf{x}$$ The non-zero vector $\mathbf{x}$ satisfying this equation is called the eige...