Spectral Theorem

Diagonalizing a matrix AA means changing into a basis where AA is diagonal.
In other words

A=PDP1A = PDP^{-1}

Each column of P must be an eigenvector since

A=PDP1    AP=PD    Apj=dipj    Ax=λxA = PDP^{-1} \implies AP = PD \iff Ap_j = d_i p_j \iff Ax = \lambda x

From now on I'll write DD as Λ\Lambda and PP as XX 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



Diagonalizing a matrix is super useful, it lets you compute matrix exponents MkM^k "instantly", and compute eMe^M to solve differential equations.