Regular matrix polynomials and homogeneous eigenpairs

Fix integers $n,m\ge 1$ and coefficient matrices $A_0,\dots ,A_m\in {\mathbb{C}}^{n\times n}$. The grade-$m$ matrix polynomial is $P(\lambda ):={\sum }_{j=0}^m{A}_j{\lambda }^j\mathrm{.}$ When $P$ is not identically zero, its actual degree is the degree of its largest nonzero scalar entry and can be smaller than the declared grade $m$; the zero polynomial has no actual degree. It is regular when the scalar polynomial $\det P(\lambda )$ is not identically zero. Write $x^{\ast }$ for conjugate transpose, $⟨u,v⟩:=v^{\ast }u$ for the Euclidean inner product, $\mathrm{\parallel }x{\mathrm{\parallel }}_2:=(x^{\ast }x{)}^{1\mathrm{/}2}$ for the Euclidean vector norm, and $\mathrm{\parallel }M{\mathrm{\parallel }}_F:=({\sum }_{r,s}\mathrm{\mid }{M}_{rs}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ for the Frobenius matrix norm.

The grade-$m$ homogeneous polynomial is $P_h(\alpha ,\beta ):={\sum }_{j=0}^m{A}_j{\alpha }^j{\beta }^{m-j}\mathrm{.}$ A projective parameter $[\alpha :\beta ]$ is a nonzero pair $(\alpha ,\beta )\in {\mathbb{C}}^2$ modulo common multiplication by a nonzero scalar. A homogeneous eigenpair is $([\alpha :\beta ],x)$ with $x\in {\mathbb{C}}^n\setminus \{0\}$ and $P_h(\alpha ,\beta )x=0$. This condition is independent of the representative because $P_h(c\alpha ,c\beta )=c^m{P}_h(\alpha ,\beta )$. If $\beta \mathrm{\ne }0$, the associated finite parameter is $\lambda =\alpha \mathrm{/}\beta$ and $P_h(\alpha ,\beta )={\beta }^{m}P(\lambda )$. The point $[1:0]$ is the parameter at infinity; there $P_h(1,0)=A_m$, so it is an eigenvalue exactly when $A_m$ is singular. For regular $P$, the algebraic multiplicity at a finite ${\lambda }_0$ is the order of the zero of $\det P$ at ${\lambda }_0$. The reversal is $P_m^{\mathrm{r}\mathrm{e}\mathrm{v}}(\mu ):={\sum }_{j=0}^m{A}_{m-j}{\mu }^j={\mu }^{m}P(1\mathrm{/}\mu )$; the algebraic multiplicity at infinity is the order of the zero of $\det P_m^{\mathrm{r}\mathrm{e}\mathrm{v}}$ at $\mu =0$. The order of a zero at $0$ is the largest integer $r$ for which ${\mu }^r$ divides the scalar polynomial.

The grade is part of the data. Padding a polynomial with zero leading coefficients leaves its finite eigenvalues unchanged but can add eigenvalues at infinity. Regularity excludes the identically singular case while allowing $A_m$ to be singular.