# Definitions Section 5.1

An **Eigenvalue** is the scalar $\lambda$ such that the solution to the matrix $A$ multiplied by the vector $x$ is a scalar multiple times the vector $x$ itself. That is, lambda is the scalar of this equation:$Ax=\lambda x$

There exists an eigenvalue if and only if there is a nontrivial solution to the equation $(A-\lambda I)x=0$

In an $n*n$ matrix there can be at most $n$ distinct eigenvalues.

In a triangular matrix, the eigenvalues are the entries on the main diagonal.

An **Eigenvector** is the vector $x$ from the equation above and corresponds to a distinct eigenvalue.

The eigenvectors corresponding to a distinct eigenvalue together with the zero vector form the **Eigenspace** of that specific eigenvalue.