This paper solves the rational noncommutative analogue of Hilbert’s 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of Hermitian matrices in its domain, then it is a sum of Hermitian squares of noncommutative rational functions. This result is a generalisation and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without Hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every matricial solution of a linear matrix inequality $L\succeq 0$ if and only if it belongs to the rational quadratic module generated by L. The essential intermediate step toward this Positivstellensatz for functions with singularities is an extension theorem for invertible evaluations of linear matrix pencils.

The characteristic polynomial ${{P}_{A}}({{x}_{0}},...,{{x}_{r}})$ of an $r$-tuple $A\,:=({{A}_{1}},...,{{A}_{r}})$ of $n\times n$-matrices is defined as

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$${{P}_{A}}({{x}_{0}},...,{{x}_{r}}):=\det ({{x}_{0}}I+{{x}_{1}}{{A}_{1}}+\ldots +{{x}_{r}}{{A}_{r}}).$$

We show that if $r\,\,3$ and $A\,:=({{A}_{1}},...,{{A}_{r}})$ is an $r$-tuple of $n\times n$-matrices in general position, then up to conjugacy, there are only finitely many $r$-tuples $A'\,:=(A_{1}^{'},...,A_{r}^{'})$ such that ${{p}_{A}}={{p}_{A'}}$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in ${{\text{P}}^{r}}$ is irreducible of dimension $(r-1){{n}^{2}}+1$.

We describe some aspects of spectral theory that involve algebraic considerations but need no analysis. Some of the important applications of the results are to the algebra of n×n matrices with entries that are polynomials or more general analytic functions.

We investigate the perturbation of the palindromic eigenvalue problem for the matrix quadratic with A0,A1∈𝒞n×n and (where or H). The perturbation of eigenvalues in the context of general matrix polynomials, palindromic pencils, (semi-Schur) anti-triangular canonical forms and differentiation is discussed.

Let A ∈ ℒ(Cn) and A1, A2 be the unique Hermitian operators such that A = A1 + i A2. The paper is concerned with the differential structure of the numerical range map nA: x ↦ ((A1x, x), (A1x, x)) and its connection with certain natural subsets of the numerical range W(A) of A. We completely characterize the various sets of critical and regular points of the map nA as well as their respective images within W(A). In particular, we show that the plane algebraic curves introduced by R. Kippenhahn appear naturally in this context. They basically coincide with the image of the critical points of nA.

Eigenvalue problems for selfadjoint quadratic operator polynomials L(λ) = Iλ2 + Bλ+ C on a Hilbert space H are considered where B, C∈ℒ(H), C >0, and |B| ≥ kI + k-l C for some k >0. It is shown that the spectrum of L(λ) is real. The distribution of eigenvalues on the real line and other spectral properties are also discussed. The arguments rely on the well-known theory of (weakly) hyperbolic operator polynomials.