The characteristic polynomial and determinant are not ad hoc constructions

There is a digraph corresponding to every square matrix over $\mathbb{C}$. We generate a recurrence relation using the Laplace expansion to calculate the characteristic, and permanent polynomials of a square matrix. Solving this recurrence relation, we found that the characteristic, and permanent polynomials can be calculated in terms of characteristic, and permanent polynomials of some specific induced subdigraphs of blocks in the digraph, respectively. Interestingly, these ...

In this paper, the correspondence between the finite dimensional representations of a simple Lie algebra and their characteristic polynomials is established, and a monoid structure on these characteristic polynomials is constructed. Furthermore, the characteristic polynomials of sl(2, C) on some classical simple Lie algebras through adjoint representations are studied, and we present some results of Borel subalgebras and parabolic subalgebras of simple Lie algebras through ch...

Can the cross product be generalized? Why are the trace and determinant so important in matrix theory? What do all the coefficients of the characteristic polynomial represent? This paper describes a technique for `doodling' equations from linear algebra that offers elegant solutions to all these questions. The doodles, known as trace diagrams, are graphs labeled by matrices which have a correspondence to multilinear functions. This correspondence permits computations in linea...

By using combinatorics, we give a new proof for the recurrence relations of the characteristic polynomial coefficients, and then we obtain an explicit expression for the generic term of the coefficient sequence, which yields the trace formulae of the Cayley-Hamilton's theorem with all coefficients explicitly given, and which implies a byproduct, a complete expression for the determinant of any finite-dimensional matrix in terms of the traces of its successive powers. And we d...

We describe a simple approach to factorize non-commutative (nc) polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear representations. To be more specific, a correspondence between factorizations of an element and upper right blocks of zeros in the system matrix (of its representation) is established. The problem is then reduced to solving a system of polynomi...

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the fundamental theorem of algebra for quaternions to polynomials with two monomials in the leading form, while showing that it fails for three.

A formula is presented for the determinant of the second additive compound of a square matrix in terms of coefficients of its characteristic polynomial. This formula can be used to make claims about the eigenvalues of polynomial matrices, with sign patterns as an important special case. A number of corollaries and applications of this formula are given.

In this short note, we give a new sufficient condition for a linear map from a product of copies of a field to endomorphisms of a finite dimensional vector space over the same field to be an algebra homomorphism. We expect that this result can be applied to study representations of higher-degree Clifford algebras and finite extensions of commutative rings.

Let $A$ be a matrix of size $n \times n$ over an algebraically closed field $F$ and $q(t)$ a monic polynomial of degree $n$. In this article, we describe the necessary and sufficient conditions of $q(t)$ so that there exists a rank one matrix $B$ such that the characteristic polynomial of $A+B$ is $q(t)$.

We review the Preparata-Sarwate algorithm, a simple $O(n^{3.5})$ method for computing the characteristic polynomial, determinant and adjugate of an $n \times n$ matrix using only ring operations together with exact divisions by small integers. The algorithm is a baby-step giant-step version of the more well-known Faddeev-Leverrier algorithm. We make a few comments about the algorithm and evaluate its performance empirically.