The characteristic polynomial and determinant are not ad hoc constructions

We reconsider the theory of Lagrange interpolation polynomials with multiple interpolation points and apply it to linear algebra. For instance, $A$ be a linear operator satisfying a degree $n$ polynomial equation $P(A)=0$. One can see that the evaluation of a meromorphic function $F$ at $A$ is equal to $Q(A)$, where $Q$ is the degree $<n$ interpolation polynomial of $F$ with the the set of interpolation points equal to the set of roots of the polynomial $P$. In particular, fo...

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal representation. Br\"and\'en has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there ar...

Let $k$ be a field of characteristic two. We prove that a non constant monic polynomial $f\in k[X]$ of degree $n$ is the minimal/characteristic polynomial of a symmetric matrix with entries in $k$ if and only if it is not the product of pairwise distinct inseparable irreducible polynomials. In this case, we prove that $f$ is the minimal polynomial of a symmetric matrix of size $n$. We also prove that any element $\alpha\in k_{alg}$ of degree $n\geq 1$ is the eigenvalue of a s...

We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer algebra techniques, minimal basis computations and matrix fraction expansion/reconstruction, and to polynomial matrix multiplication. Such reductions eventually imply that all these problems can be solved in about the same amount of time as pol...

Let $\mathbb{H}$ be a field with $\mathbb{Q}\subset\mathbb{H}\subset\mathbb{C}$, and let $p(\lambda)$ be a polynomial in $\mathbb{H}[\lambda]$, and let $A\in\mathbb{H}^{n\times n}$ be nonderogatory. In this paper we consider the problem of finding a solution $X\in\mathbb{H}^{n\times n}$ to $p(X)=A$. A necessary condition for this to be possible is already known from a paper by M.P. Drazin. Under an additional condition we provide an explicit construction of such solutions. Th...

In this paper, we investigate characteristic polynomials of matrices in min-plus algebra. Eigenvalues of min-plus matrices are known to be the minimum roots of the characteristic polynomials based on tropical determinants which are designed from emulating standard determinants. Moreover, minimum roots of characteristic polynomials have a close relationship to graphs associated with min-plus matrices consisting of vertices and directed edges with weights. The literature has ye...

The centralizer of an endomorphism of a finite dimensional vector space is known when the endomorphism is nonderogatory or when its minimal polynomial splits over the field. It is also known for the real Jordan canonical form. In this paper we characterize the centralizer of endomorphisms over arbitrary fields for whatever minimal polynomial, and compute its dimension. The result is obtained via generalized Jordan canonical forms (for separable and non separable minimal polyn...

The coefficient algebra of a finite-dimensional Lie algebra with respect to a faithful representation is defined as the subalgebra generated by all coefficients of the corresponding characteristic polynomial. We establish a connection between classical invariant theory and the coefficient algebras of finite-dimensional complex Lie algebras. Specifically, we prove that with respect to any symmetric power of the standard representation: (1) the coefficient algebra of the upper ...

The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by attaching them some quadratic forms.

This is a detailed survey -- with rigorous and self-contained proofs -- of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is entirely expository (and written to a large extent as a repository for folklore proofs); no new results (and few, if any, new proofs) appear.