User:IssaRice/Linear algebra/Classification of operators

Let $V$ be a finite-dimensional inner product space, and let $T:V\to V$ be a linear transformation. Each row in the table below says the same thing.

Operator name	Description in terms of eigenvectors	Description in terms of diagonalizability
$T$ is diagonalizable	There exists a basis of $V$ consisting of eigenvectors of $T$	$T$ is diagonalizable (there exists a basis $\beta$ of $V$ with respect to which $[T{]}_{\beta }^{\beta }$ is a diagonal matrix)
$T$ is normal	There exists an orthonormal basis of $V$ consisting of eigenvectors of $T$	$T$ is diagonalizable using an orthonormal basis
$T$ self-adjoint ($T$ is Hermitian)	There exists an orthonormal basis of $V$ consisting of eigenvectors of $T$ with real eigenvalues	$T$ is diagonalizable using an orthonormal basis and the diagonal entries are all real
$T$ is an isometry
$T$ is positive