Diagonalizing a matrix $A$ means changing into a basis where $A$ is diagonal.
In other words
$A = PDP^{-1}$
Each column of P must be an eigenvector since
$A = PDP^{-1} \implies AP = PD \iff Ap_j = d_i p_j \iff Ax = \lambda x$
From now on I'll write $D$ as $\Lambda$ and $P$ as $X$ to signify the connection with eigenvectors.
When can a matrix be diagonalized? (*)
In other words, is there a full set of eigenvectors that spans the space?
Symmetric matrices have real eigenvalues
[x] Show that diagonalizing is identical to finding a full set of eigenvectors.
[ ] Explain what diagonalizing is, why we want to do it
[ ] Symmetric case $A = A^T \implies A = U\Lambda X^T$ real matrices
[ ] General case $M^TM = MM^T \implies M = U\Lambda U^T$
[ ] Prove $M = M^T \implies \lambda_i \in \mathbb R$
Diagonalizing a matrix is super useful, it lets you compute matrix exponents $M^k$ "instantly", and compute $e^M$ to solve differential equations.