  We introduce filling families with matrix diagonalization as a refinement of
the work by R{\o}rdam and the first named author. As an application we improve
a result on local pure infiniteness and show that the minimal tensor product of
a strongly purely infinite $C^*$-algebra and a exact $C^*$-algebra is again
strongly purely infinite. Our results also yield a sufficient criterion for the
strong pure infiniteness of crossed products $A\rtimes_\varphi \mathbb{N}$ by
an endomorphism $\varphi$ of $A$ (cf. Theorem 7.6). Our work confirms that the
special class of nuclear Cuntz-Pimsner algebras constructed by Harnisch and the
first named author consist of strongly purely infinite $C^*$-algebras, and thus
absorb $\mathcal{O}_\infty$ tensorially.
