Generalized Unitary Matrix

Generalized Unitary Matrix

A unitary matrix is an $N\times N$ (square) complex-valued matrix
$\mathbf{A}$ satisfying:
$\mathbf{AA^\dagger} = \mathbf{I}_N$
where $\mathbf{I}_N$ is the identity matrix.
Suppose an $N\times N$ (square) complex-valued matrix $\mathbf{B}$ satisfies:
$\mathbf{BB^\dagger} = \mathbf{D}_N$
where $\mathbf{D}_N$ is some diagonal matrix.
What can be said about $\mathbf{B}$ ?
I am particularly interested in the condition number or similar properties
reflecting a degree of orthogonality or isometry property.