A Simple Proof of the Polar Decomposition Theorem

Marden's theorem characterizes the critical points of complex polynomials of degree 3 in a nice geometrical way. Our proof of the theorem is based directly on the defining property of ellipses.

Numerical approximation of the product of the square root of a matrix with a vector

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

The use of a matrix Riccati equation to establish sufficiency theorems in the calculus of variations is well known (see [3], e.g.). In this note we extend the method to give an elementary proof of the Morse index theorem, thereby eliminating the ad hoc subdivisions of the classical proof (see [2], [4], or [5], e.g.). Although we consider only the simplest form of the index theorem, the technique can easily be adapted to generalizations such as that of [N]. Let E be a euclidean space, G the linear space of broken CIO maps u: [0, T]-->E for some closed interval [0, T], and H the subspace of those uEG such that u(0) =u(T) =0. Let P be a given Coo map of [0, T] into the selfadjoint linear transformations of E, and let U: [0, T]->HomR(E, E) be the unique Coo map satisfying U(O) =0, U' (0) = I, and +P U= 0; U can also be regarded as a linear transformation of G into itself which is stable on the subspace H. The multiplicity of any t& (O, T] is the nullity of U(t), and t is a focal point whenever it has positive multiplicity. The index form I on HXH is defined by I(u, v)-=ft 'o[(u', v')-(Pu, v)]dt, and an inner product J on HXH is defined by J(u, v) =fT0(u, v)dt, where (, ) is in both cases the inner product on E XE; for convenience we set I(u, u) = I(u) and J(u, u) =J(u). The index i(I) is the dimension of any maximal subspace of H on which I is negative definite, and the nullity n(I) is the dimension of the subspace of those u CH such that I(u, v)=0 for all vEH. The first portion of our proof of the index theorem is standard, depending on the following classical lemma:

FollowingWelters, we describe howthe use of a different degeneration from that considered by Eisenbud and Harris leads to a simple and characteristic-independent proof of the Brill-Noether theorem using limit linear series. As suggested by the degeneration, we prove an extended version of the theorem allowing for imposed ramification at up to two points.Although experts in the field have long been aware of themain ideas, we address some technical issues which arise in proving the full version of the theorem.

As an application of a sharpL2extension theorem for holomorphic functions in Guan and Zhou, a stability theorem for the boundary asymptotics of the Bergman kernel is proved. An alternate proof of the extension theorem is given, too. It is a simplified proof in the sense that it is free from ordinary differential equations.

In this paper a very easy proof of the fundamental theorem of finite linear spaces is given.

We describe the formalization using the HOL Light theorem prover of Dirichlet's theorem on primes in arithmetic progression. The proof turned out to be more straightforward than expected, but this depended on a careful choice of an informal proof to use as a starting-point. The goal of this paper iis twofold. First we describe a simple and efficient proof of the theorem informally, which iis otherwise difficult to find in one self-contained place at an elementary level. We also describe its, largely routine, HOL Light formalization, a task that took only a few days.

: Edmonds has given a complicated algorithmic proof of a theorem characterizing directed graphs that contain k edge-disjoint branchings having specified root sets. Tarjan has described a conceptually simple and good algorithm for finding such branchings when they exist. Tarjan's algorithm is based on a lemma implicit in Edmonds' results. A simple direct proof of this lemma is given, thereby providing a simpler proof of Edmonds' theorem and a simpler proof that Tarjan's algorithm works.

This paper develops a preconditioning theory for the standard optimal control problem. Preconditioning is the execution of a coordinate transformation in order to increase the convergence rate of gradient based numerical algorithms. The transforming operator employed for this purpose is the square root of a symmetric positive definite matrix which is called the preconditioner. It was shown in [l] that the optimal preconditioner for nonlinear constrained mathematical programming problems is a matrix, denoted as 'H+, which is the positive definite reflection of the Hessian matrix 'H. By positive definite reflection, it is meant that matrix whose eigenvectors are identical to those of the Hessian and whose eigenvalues are the moduli of the latter's eigenvalues. It was demonstrated that the preconditioner 'H+ produces a transformed Hessian with a condition number of unity; it was further shown how H+ can be efficiently constructed from a singular value decomposition of 7-t. Since the Hessian is a function of the optimizing variables in a nonlinear problem, periodic application of the preconditioner is required for best results. Here, this preconditioning theory is extended to the standard optimal control problem using the direct approach of transcription. The optimal preconditioner, as well as suboptimal alternatives, are presented based on the Hessian of the resulting finite dimensional problem. Additionally, by allowing the discretization level to approach zero, the infinite-dimensional Hessian of the optimal control problem is obtained, which turns out to be a sparse matrix-valued hybrid surface consisting of ridges and discrete points. This is a nice geometrical result which provides great insight into the optimal control problem. It is shown for completeness, that as At -+ 0, the necessary conditions and the second differential of the mathematical programming problem converge to the necessary conditions and the second variation of the control problem. It is further shown that the second variation of the control problem has the form ;AZ~ 'HAZ, where the Hessian is a hybrid surface and Ar is an infinite-dimensional vector. For the second variation to be positive, the Hessian sur'Associate Fellow, AIAA 'Member, AIAA This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States. face must be positive definite with respect to surface multiplication, surface multiplication being the infinite-dimensional extension of matrix multiplication. Finally, it is noted that the extended theory is also valid in the indirect approach to optimal control provided a new integration implementation technique is employed, as shown in the separate work [2].

The usual solution methods of discretised partial differential equations are based exclusively on matrix-vector multiplications as basis operation. On the one hand, this is supported by the use of sparse matrices (cf. §1.3.2.5); on the other hand, one tries to apply fast iterative methods (e.g., the multigrid method [124], [119, §12]) whose basic steps are matrix-vector multiplications. Krylov methods are based on the same concept. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation1 and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. The $$\mathcal{H}$$ -matrix arithmetic allows the solution of these matrix equations efficiently. Here, the use of hierarchical matrix operations and matrix-valued functions is only one part of the solution method. Another essential fact is that the solution $$ X \in \mathbb{R}^{I \times I}(n==\!\!\!\!\!/ \!\!\!\!/I)$$ can be replaced by an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ . If one considers the equation $$f(X)\;=\;0$$ as a system of n 2 equation for the n 2 components of X, even an optimal solution method has complexity $$\mathcal{O}(n^2)$$ , since this is linear complexity in a number of unknowns (cf. Remark 1.1). Using traditional techniques, the solution of large-scale matrix equations is not feasible. Only an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ with $$\mathcal{O}(n\; \mathrm{log}^*n)$$ data instead of n 2 admits a solution with a cost almost linear with respect to n. Section 15.1 introduces Lyapunov and Sylvester equations and discusses their solution. In Section 15.2 we consider quadratic Riccati equation. An interesting approach uses the matrix version of the sign function from §14.1.1. General nonlinear matrix equations may be solved iteratively by Newton’s method or related methods (cf. Section 15.3). As an example, computing the square root of a positive definite matrix is described in §15.3.1. The influence of the truncation error introduced by H-matrix arithmetic is investigated in Section 15.3.

This paper is triggered by the preprint [P. Jain, C. Jin, S.M. Kakade, and P. Netrapalli. Computing matrix squareroot via non convex local search. Preprint, arXiv:1507.05854, 2015.], which analyzes gradient-descent for computing the square root of a positive definite matrix. Contrary to claims of Jain et al., the author’s experiments reveal that Newton-like methods compute matrix square roots rapidly and reliably, even for highly ill-conditioned matrices and without requiring com-mutativity. The author observes that gradient-descent converges very slowly primarily due to tiny step-sizes and ill-conditioning. The paper derives an alternative first-order method based on geodesic convexity; this method admits a transparent convergence analysis (< 1 page), attains linear rate, and displays reliable convergence even for rank deficient problems. Though superior to gradient-descent, ultimately this method is also outperformed by a well-known scaled Newton method. Nevertheless, the primary value of the paper is conceptual: it shows that for deriving gradient based methods for the matrix square root, the manifold geometric view of positive definite matrices can be much more advantageous than the Euclidean view.

In this note I consider a simple proof of Arrow's Impossibility Theorem (Arrow 1963). I start with the case of three individuals who have preferences on three alternatives. In this special case there are 133=2197 possible combinations of the three individuals' rational preferences. However, by considering the subset of linear preferences, and employing the full strength of the IIA axiom, I reduce the number of cases necessary to completely describe the SWF to a small number, allowing an elementary proof suitable for most undergraduate students.

A simple proof of the Cayley-Hamilton theorem for the case of two-dimensional systems is presented. The proof is based on the recursive two-dimensional Leverrier-Faddeeva algorithm.

We give a simple proof of the existence of a complex eigenvalue of real or complex square matrices with Linear Algebra accessible to students beginning a first course on the subject, namely using just the Cauchy–Schwarz inequality in , linear independence and the matrix product, together with the Weierstrass extreme value theorem. It yields an equally simple proof of the Fundamental Theorem of Algebra with Linear Algebra.