Related papers: A direct approach to the rational normal form
In this note, a simple proof Jordan normal form and rational form of matrices over a field is given.
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
In this paper, we derive the quadratic formula as a consequence of constructively proving the existence of standard and factored forms for general form real quadratic functions. Emphasis is put on connections to graphing of corresponding…
We give a short proof -- not relying on ideal classes or the geometry of numbers -- of a known criterion for quadratic orders to possess unique factorization.
We give an elementary proof of a recent result by Fishman, Kleinbock, Merrill and Simmons about rational points on quadratic surfaces.
This note contains a short proof of a classical result: any rational symplectic matrix can be put in diagonal form after right and left multiplication by integral symplectic matrices.
We give a canonical form for a complex matrix, whose square is normal, under transformations of unitary similarity as well as a canonical form for a real matrix, whose square is normal, under transformations of orthogonal similarity.
This note presents a short, transparent proof of the theorem that every Euclidean quadratic form over a normed integral domain is an Aubry-Davenport-Cassels form. The theorem, as formulated in the note, allows besides quadratic terms also…
This note documents the specification of normal forms in cubical type theory. The definition is already present in the proof of normalization for cubical type theory, but we present it in a more traditional style explicitly for reference.
Matrices over the ring of formal power series are considered. Normal forms with respect to various sub-groups of the two-sided transformations are constructed. The construction is based on the special property of the action: it induces a…
Let $V$ be a finite dimensional vector space over a field $\mathrm{k}$ of characteristic $0$. Let $A$ be a linear mapping of $V$ into itself. This paper gives a normal form for $A$, which gives a better description of the structure of $A$…
In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.
We compute the genus of a rational quadratic form in terms of the K-theory of a C*-algebra attached to the adelic orthogonal group of the form. As a corollary, one gets a higher composition law for the rational quadratic forms. As an…
For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.
We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is…
We prove a result on the existence of linear forms of a given Diophantine type.
We prove that the orbit closure of the determinant is not normal. A similar result is obtained for the orbit closure of the permanent multiplied by a power of a linear form.
In this paper we show that Cardanos formula for the solution of cubic equations can be reduced to expressions involving only square roots if the real root is rational.
This will is an expository description of quadratic rational maps. Sections 2 through 6 are concerned with the geometry and topology of such maps. Sections 7--10 survey of some topics from the dynamics of quadratic rational maps. There are…
We show by finding an explicit parametrization that a 4th degree surface which arises as a necessary condition for the existence of a perfect cuboid is a rational surface, i.e. birationally equivalent over $\mathbb Q$ to a plane.