Operator Trigonometry Research Articles

Hermitian matrices: Spectral coupling, plane geometry/trigonometry and optimisation

The paper presents the information processing that can be performed by a general hermitian matrix when two of its distinct eigenvalues are coupled, such as λ<λ′, instead of considering only one eigenvalue as traditional spectral theory does. Setting a=λ+λ′2≠0 and e=λ′−λ2>0, the information is delivered in geometric form, both metric and trigonometric, associated with various right-angled triangles exhibiting optimality properties quantified as ratios or product of |a| and e. The potential optimisation has a triple nature which offers two possibilities: in the case λλ′>0 they are characterised by e|a| and |a|e and in the case λλ′<0 by |a|e and |a|e. This nature is revealed by a key generalisation to indefinite matrices over R or C of Gustafson's operator trigonometry.

On my Min-Max Theorem (1968) and its consequences

Central to the origins of my operator trigonometry, a theory in which I initiated the concepts of antieigenvalues and antieigenvectors, is my 1968 Min-Max Theorem. I will discuss its motivation, proof, and consequences. Special emphasis will be given here to a new view that sin φ(A) may be viewed as a general optimum which encompasses many other optima of individual interest.

Operator trigonometry of multivariate finance

We inquire into an operator-trigonometric analysis of certain multi-asset financial pricing models. Our goal is to provide a new geometric point of view for the understanding and analysis of such financial instruments. Among those instruments which we examine are quantos for currency hedging, spread options for multi-asset pricing, portfolio rebalancing under stochastic interest rates, Black–Scholes volatility models, and risk measures.

Operator trigonometry of Hotelling correlation, Frobenius condition, Penrose twistor

Three new applications of the author’s operator trigonometry are presented. These take place in three diverse mathematical domains: matrix statistics, numerical analysis, and theoretical physics. The goal is to continue to develop and expand the scope of the operator trigonometry.

Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics

Let A be p×p positive definite matrix. A p-vector x such that Ax=λx is called an eigenvector with the associated with eigenvalue λ. Equivalent characterizations are: (i) cosθ=1, where θ is the angle between x and Ax. (ii) (x'Ax) -1 =xA -1 x. (iii) cosΦ=1, where φ is the angle between A ½ x and A -½ x. We ask the question what is x such that cosθ as defined in (i) is a minimum or the angle of separation between x and Ax is a maximum. Such a vector is called an anti-eigenvector and cosθ an anti-eigenvalue of A. This is the basis of operator trigonometry developed by K. Gustafson and P.D.K.M. Rao (1997), Numerical Range: The Field of Values of Linear Operators and Matrices, Springer. We may define a measure of departure from condition (ii) as min[(x'Ax)(x'A -1 x)] -1 which gives the same anti-eigenvalue. The same result holds if the maximum of the angle Φ between A ½ x and A -½ x as in condition (iii) is sought. We define a hierarchical series of anti-eigenvalues, and also consider optimization problems associated with measures of separation between an r(<p) dimensional subspace S and its transform AS. Similar problems are considered for a general matrix A and its singular values leading to anti-singular values. Other possible definitions of anti-eigen and anti-singular values, and applications to problems in statistics will be presented.

Bell and Zeno

Bell's inequalities and related inequalities of Wigner, Clauser–Horne–Shimony–Holt, Accardi–Fedullo, Gudder–Zanghi, Herbert–Peres, Khrennikov, others, are shown to be contained within a general operator trigonometry developed by this author starting in 1967. These inequalities are improved here to useful quantum spin correlation identities. Secondly, the Zeno problems from quantum measurement theory are traced from early work by this author starting in 1974, to the present. A Zeno Alternative that stresses domain-theoretic properties as essential to distinguishing reversible from irreversible quantum evolutions is presented.

Operator trigonometry of preconditioning, domain decomposition, sparse approximate inverses, successive overrelaxation, minimum residual schemes

AbstractThis paper extends the relationship between the author's operator trigonometry and convergence rates and other properties of important iterative methods. A new interesting trigonometric preconditioning lemma is given. The general relationship between domain decomposition methods and the operator trigonometry is established. A new basic conceptual link between sparse approximate inverse algorithms and the operator trigonometry is observed. A new underlying fundamental inherent trigonometry of the classical successive over‐relaxation scheme is exposed. Some improved trigonometric interpretations of minimum residual schemes are mentioned. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Operator trigonometry of statistics and econometrics

A new and useful geometric point of view for the understanding and analysis of certain matrix methods as they are used in statistics and econometrics is presented. Applications to statistical efficiency, parameter estimation, and correlation theory are given. In particular we show that worst case relative least squares efficiency, although achieved by maximally inefficient regressors, is also achieved by maximal covariance matrix turning vectors. Also we elaborate geometrically a commutator trace efficiency result of P. Bloomfield and G.S. Watson [Biometrika 62 (1975) 121]. Well-established Lagrange multiplier methods for constrained optimizations are compared to use of an Euler equation from the new geometric theory.

Probability, geometry, and irreversibility in quantum mechanics

The main conclusion of this paper is that the Bell–Wigner–Accardi theory of quantum probabilities in spin systems may be placed within the general operator trigonometry developed independently by this author about 30 years ago. The use of the Grammian from the operator trigonometry simplifies and clarifies the analysis of Wigner. A general triangle inequality from the operator trigonometry clarifies and generalizes the analysis of Accardi. The statistical meaning of the complex numbers in quantum mechanics is seen to be that of the natural geometry of the operator trigonometry. A new connection of the operator trigonometry to CP symmetry violation is established.

An extended operator trigonometry

The operator trigonometry of symmetric positive definite (SPD) matrices is extended to arbitrary invertible matrices A and to arbitrary invertible bounded operators A on a Hilbert space. Some background and motivation for these results is provided.

QUANTUM TRIGONOMETRY

Our purpose here is to establish that a general operator trigonometry created by us 30 years ago provides a natural geometry and useful tool for certain situations occurring in quantum probability. In particular we will embed the Bell–Wigner–Accardi theory of quantum spin systems into this operator trigonometry. We will also establish a new connection of the operator trigonometry to CP symmetry violation. We begin with a hopefully helpful brief history starting with the famous Einstein–Podolsky–Rosen gedankenexperiment paper (1935). We close with a short discussion of possible future developments.

The geometrical meaning of the Kantorovich–Wielandt inequalities

The Kantorovich–Wielandt angle θ( A) and the author's operator angle φ( A) are related by cos φ(A 2)= sin θ(A) . Here A is an arbitrary symmetric positive definite (SPD) matrix. The relationship of these two different geometrical perspectives is discussed. An extension to arbitrary nonsingular matrices A is given. A related four-way relationship with the operator trigonometry, strengthened CBS constants, and domain decomposition methods is noted.

Operator trigonometry of the model problem

Operator trigonometry of the model problem

Operator Trigonometry of Linear Systems

Operator trigonometry, which originated in problems of functional analysis in the late 1960's, is applied to iterative methods for the solution of linear systems Ax = b. The operator trigonometry provides a new theory for the understanding and analysis of such methods. For example, the famous Kantorovich error bound for gradient algorithms is now seen to be inherently trigonometric. The basic Richardson scheme also turns out to be fundamentally trigonometric. Stability of control is also seen trigonometrically.

Operator trigonometry of iterative methods

Operator trigonometry of iterative methods

Operator trigonometry of iterative methods

Operator trigonometry of iterative methods

Operator trigonometry

The purpose of the present paper is to survey, from a historical perspective and including some new results, a theory which I will call operator trigonometry. This theory, which is little known, is closely associated with the numerical range W(A) of an operator A. Among the new results is a beautiful connection to numerical linear algebra, in which gradient descent and conjugate gradient convergence rates are shown to be trigonometric.