Diagonalizing a matrix means changing into a basis where is diagonal.
In other words
Each column of P must be an eigenvector since
From now on I'll write as and as 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 real matrices
[ ] General case
[ ] Prove
Diagonalizing a matrix is super useful, it lets you compute matrix exponents "instantly", and compute to solve differential equations.