Related papers: D\'eterminant d'une matrice carr\'ee
This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The main result in…
This brief note concerns the invertibility of certain alternant matrices. In particular those that consisting of polynomials and products of polynomials and logarithms are shown to be invertible under appropriate conditions on the degrees…
We characterize the finite groups of minimal order that admit an irreducible complex character of degree $p$ or $p^2$, where $p$ is a prime.
We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of…
In this paper we generalize the notion of discriminant for symmetric matrices and some results about it to the case of symmetric spaces.
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
All matrices we consider have entries in a fixed algebraically closed field $K$. A minor of a square matrix is principal means it is defined by the same row and column indices. We study the ideal generated by size $t$ principal minors of a…
The Minor problem, namely the study of the spectrum of a principal submatrix of a Hermitian matrix taken at random on its orbit under conjugation, is revisited, with emphasis on the use of orbital integrals and on the connection with…
We evaluate determinants of "spiral" matrices, which are matrices in which entries are spiralling from the centre of the matrices towards the outside, with prescribed increments from one entry to the next depending on whether one moves…
We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our…
In our article we consider some algebraical methods which may be useful in some inverse spectral problems. The reconstraction of the matrix from its minors is considered.
A compound determinant identity for minors of rectangular matrices is established. As an application, we derive Vandermonde type determinant formulae for classical group characters.
In the following short paper we list some useful results concerning determinants and inverses of matrices. First we show, how to calculate determinants of $d \times d$ matrices, if their traces are known. As a next step $4 \times 4$…
The work considers an equivalence relation in the set of all $n\times m$ matrices with entries in the set $[p]=\{ 0,1,\ldots , p-1 \}$. In each element of the factor-set generated by this relation, we define the concept of canonical matrix,…
Square matrices of the form $\widetilde{\mathbf{A}} =\mathbf{A} + \mathbf{e}D \mathbf{f}^*$ are considered. An explicit expression for the inverse is given, provided $\widetilde{\mathbf{A}}$ and $D$ are invertible with…
We determine sufficient conditions for certain classes of $(n+k) \times n$ matrices $E$ to have all order-$n$ minors to be nonzero. For a special class of $(n+1) \times n$ matrices $E,$ we give the formula for the order-$n$ minors. As an…
The purpose of this note is to give explicit criteria to determine whether a real generalized Cartan matrix is of finite type, affine type or of hyperbolic type by considering the principal minors and the inverse of the matrix. In…
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, ``conformal'' (multicomponent) and Kontsevich models are considered in some detail, together with the…
The iteration of rational maps is well-understood in dimension 1 but less so in higher dimensions. We study some maps on spaces of matrices which present a weak complexity with respect to the ring structure. First we give some properties of…
For given real or complex $m \times n$ data matrices $X$, $Y$, we investigate when there is a matrix $A$ such that $AX = Y$, and $A$ is invertible, Hermitian, positive (semi)definite, unitary, an orthogonal projection, a reflection, complex…