The conditioning of block Kronecker $\ell$-ifications of matrix polynomials

A strong $\ell$-ification of a matrix polynomial $P(\lambda)=\sum A_i\lambda^i$ of degree $d$ is a matrix polynomial $\mathcal{L}(\lambda)$ of degree $\ell$ having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as $P(\lambda)$. Strong $\ell$-ifications can be used to transform the polynomial eigenvalue problem associated with $P(\lambda)$ into an equivalent polynomial eigenvalue problem associated with a larger matrix polynomial $\mathcal{L}(\lambda)$ of lower degree. Typically $\ell=1$ and, in this case, $\mathcal{L}(\lambda)$ receives the name of strong linearization. However, there exist some situations, e.g., the preservation of algebraic structures, in which it is more convenient to replace strong linearizations by other low degree matrix polynomials. In this work, we investigate the eigenvalue conditioning of $\ell$-ifications from a family of matrix polynomials recently identified and studied by Dopico, P\'erez and Van Dooren, the so-called block Kronecker companion forms. We compare the conditioning of these $\ell$-ifications with that of the matrix polynomial $P(\lambda)$, and show that they are about as well conditioned as the original polynomial, provided we scale $P(\lambda)$ so that $\max\{\|A_i\|_2\}=1$, and the quantity $\min\{\|A_0\|_2,\|A_d\|_2\}$ is not too small. Moreover, under the scaling assumption $\max\{\|A_i\|_2\}=1$, we show that any block Kronecker companion form, regardless of its degree or block structure, is about as well-conditioned as the well-known Frobenius companion forms. Our theory is illustrated by numerical examples.