Theorems from Diagonalisation

Eigenvectors belonging to distinct eigenvalues are linearly independent. In particular, an $n\times n$ matrix with $n$ distinct eigenvalues has $n$ linearly independent eigenvectors.

An $n\times n$ matrix over $F$ is diagonalisable over $F$ if and only if it has $n$ linearly independent eigenvectors in $F^n$.

An $n\times n$ matrix $A$ is invertible if and only if $0$ is not an eigenvalue of $A$.