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Let H be an infinite-dimensional complex Hilbert space and let G ∞ (H) be the set of all closed subspaces of H whose dimension and codimension both are infinite. We investigate (not necessarily surjective) transformations of G ∞ (H) sending every pair of subspaces to an equivalent pair of subspaces; two pairs of subspaces are equivalent if there is a linear isometry sending one of these pairs to the other. Let f be such a transformation. We show that there is up to a scalar multiple a unique linear or conjugate-linear isometry L : H → H such that for every X ∈ G ∞ (H) the image f (X) is the sum of L (X) and a certain closed subspace O (X) orthogonal to the range of L. In the case when H is separable, we give the following sufficient condition to assert that f is induced by a linear or conjugate-linear isometry: if O (X) = 0 for a certain X ∈ G ∞ (H) , then O (Y) = 0 for all Y ∈ G ∞ (H). [ABSTRACT FROM AUTHOR]

Let n be a positive integer and H a Hilbert space. The description of the general form of bijective maps on the set of n-dimensional subspaces of H preserving the maximal principal angle has been obtained recently. This is a generalization of Wigner's unitary-antiunitary theorem. In this paper we will obtain another extension of Wigner's theorem in which the maximal principal angle is replaced by the minimal one. Moreover, in this case we do not need the bijectivity assumption. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of principal angles between subspaces of the same signature in a (real, complex or quaternionic) pseudo-euclidean space of index 1. We show that these determine the relative position of an important class of pairs of hyperbolic and elliptic subspaces (Theorems 3.9 and 4.10). Also, as an application, we will see that these angles can be used to study the relative position of pairs of an important class of totally geodesic submanifolds of the (real, complex or quaternionic) hyperbolic space (Theorem 6.3). As a consequence we obtain a nice interpretation of these principal angles as geometric invariants of the hyperbolic space. [ABSTRACT FROM AUTHOR]

In this work we revisit the convergence analysis of the Subspace Iteration Method (SIM) for the computation of approximations of a matrix A by matrices of rank h. Typically, the analysis of convergence of these low-rank approximations has been obtained by first estimating the (angular) distance between the subspaces produced by the SIM and the dominant subspaces of A. It has been noticed that this approach leads to upper bounds that overestimate the approximation error in case the hth singular value of A lies in a cluster of singular values. To overcome this difficulty we introduce a substitute for dominant subspaces, which we call admissible subspaces. We develop a proximity analysis of subspaces produced by the SIM to admissible subspaces; in turn, this analysis allows us to obtain novel estimates for the approximation error by low-rank matrices obtained by the implementation of the deterministic SIM. Our results apply in the case when the hth singular value of A belongs to a cluster of singular values. Indeed, our approach allows us to consider the case when the hth and the (h + 1) st singular values of A coincide, which does not seem to be covered by previous works in the deterministic setting. [ABSTRACT FROM AUTHOR]

We study the well-known methods of alternating and simultaneous projections when applied to two nonorthogonal linear subspaces of a real Euclidean space. Assuming that both of the methods have a common starting point chosen from either one of the subspaces, we show that the method of alternating projections converges significantly faster than the method of simultaneous projections. On the other hand, we provide examples of subspaces and starting points, where the method of simultaneous projections outperforms the method of alternating projections. [ABSTRACT FROM AUTHOR]

Convergence analysis of block iterative solvers for Hermitian eigenvalue problems and closely related research on properties of matrix-based signal filters are challenging and are attracting increased attention due to their recent applications in spectral data clustering and graph-based signal processing. We combine majorization-based techniques pioneered for investigating the Rayleigh--Ritz method in [A. V. Knyazev and M. E. Argentati, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1521-1537] with tools of classical analysis of the block power method by Rutishauser [Numer. Math., 13 (1969), pp. 4-13] to derive sharp convergence rate bounds of abstract block iterations, wherein tuples of tangents of principal angles or relative errors of Ritz values are bounded using majorization in terms of arranged partial sums and tuples of convergence factors. Our novel bounds are robust in the presence of clusters of eigenvalues, improve previous results, and are applicable to most known block iterative solvers and matrix-based filters, e.g., to block power, Chebyshev, and Lanczos methods combined with polynomial filtering. The sharpness of our bounds is fundamental, implying that the bounds cannot be improved without further assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we study some parameters which determine the relative position of a pair of quaternionic subspaces and a pair of totally geodesic submanifolds in the real hyperbolic space. We extend some results about principal angles between two real and complex subspaces for quaternionic subspaces. We also prove that the distance and the principal angles of certain tangent spaces determine the congruence class of a non-asymptotic pair of totally geodesic submanifolds of the real hyperbolic space under the action of its isometry group. [ABSTRACT FROM AUTHOR]

We introduce discrepancy values , quantities inspired by the notion of spectral spread of Hermitian matrices. We define them as the discrepancy between two consecutive Ky-Fan-like seminorms. As a result, discrepancy values share many properties with singular values and eigenvalues, yet are substantially different to merit their own study. We describe key properties of discrepancy values, and establish tools such as representation theorems, majorization inequalities, convex formulations, etc., for working with them. As an important application, we illustrate the role of discrepancy values in deriving tight bounds on the norms of commutators. [ABSTRACT FROM AUTHOR]

Let A be a d × d complex self-adjoint matrix, let X, Y ? Cd be k-dimensional subspaces, and let X be a d×k complex matrix whose columns form an orthonormal basis of X; that is, X is an isometry whose range is the subspace X. We construct a d × k complex matrix Yr whose columns form an orthonormal basis of Y and obtain sharp upper bounds for the singular values s(X*AX - Y * r AYr) in terms of submajorization relations involving the principal angles between X and Y and the spectral spread of A. We apply these results to obtain sharp upper bounds for the absolute variation of the Ritz values of A associated with the subspaces X and Y that partially confirm conjectures by Knyazev and Argentati. [ABSTRACT FROM AUTHOR]

Let A be an n × n complex Hermitian matrix and let λ (A) = (λ 1 , ... , λ n) ∈ R n denote the eigenvalues of A , counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of A , denoted Spr + (A) , given by Spr + (A) = (λ 1 − λ n , λ 2 − λ n − 1 , ... , λ k − λ n − k + 1) ∈ R k , where k = [ n / 2 ] (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of A , that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao's inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan's inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix. [ABSTRACT FROM AUTHOR]

We present a novel and computationally simple solution to the problem of detecting the number of signals, which is applicable to both white and colored noise, and to a very small number of samples. The solution is based on a novel and non-asymptotic goodness-of-fit metric, referred to as signal subspace matching (SSM), which is aimed at matching a model-based signal subspace to its sampled-data-based counterpart. We form a set of hypothesized signal subspace models, with the k-th model being a projection matrix composed of the k leading eigenvectors of the sample-covariance matrix. This set of hypothesized models is compared to their sampled-data-based counterpart – a projection matrix constructed from the sampled data – via the SSM metric, and the model minimizing this metric is selected. We show that this solution involves the principal angles between the column span of the model and the column span of the model. We prove the consistency of this solution for the high signal-to-noise-ratio limit and for the large-sample limit. The large-sample consistency is shown to be conditioned on the signal-to-noise ratio (SNR) being higher than a a certain threshold. Simulation results, demonstrating the performance of the solution for both colored and white noise, are included. [ABSTRACT FROM AUTHOR]

We present a novel solution to the problem of estimating the array response of the signal of interest (SOI) in case it is constrained to lie in a known subspace, aimed at coping with model errors in the known subspace. The solution is based on a novel formulation of the problem, targeted at matching the error-contaminated model-based signal subspace to its sampled-data counterpart. The solution turns out to minimize the angle between these two subspaces, which is intuitively very pleasing. We solve the problem for three different characterization of the spatial interference: (i) the spatial interference is known, (ii) the spatial interference is unknown, and (iii) the spatial interference is constrained to lie in a known subspace. We present a closed-form solution for the first case and iterative solutions for the other two cases. Based on these solutions, we derive the corresponding estimators for the SOI’s waveform and their signal-to-interference+noise ratio (SINR). Simulation results, demonstrating the superiority of the derived solutions over the corresponding deterministic maximum likelihood (DML) solutions, are included. [ABSTRACT FROM AUTHOR]

We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean spaces the convergence of the alternating method is not determined by the principal angles between the subspaces involved. In the second part, we investigate the properties of the Oppenheim angle between two linear projections. We discuss, in particular, the question of existence and uniqueness of "consistency projections" in this context. [ABSTRACT FROM AUTHOR]

A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and Zhu, which extend several known results for one dimensional subspaces to arbitrary subspaces. In addition, we improve Nakatsukasa's version of the tan theorem of Davis and Kahan. As a consequence, we obtain new quadratic a posteriori bounds for the absolute change in Ritz values. [ABSTRACT FROM AUTHOR]

Abstract Background Essential Dynamics (ED) is a common application of principal component analysis (PCA) to extract biologically relevant motions from atomic trajectories of proteins. Covariance and correlation based PCA are two common approaches to determine PCA modes (eigenvectors) and their eigenvalues. Protein dynamics can be characterized in terms of Cartesian coordinates or internal distance pairs. In understanding protein dynamics, a comparison of trajectories taken from a set of proteins for similarity assessment provides insight into conserved mechanisms. Comprehensive software is needed to facilitate comparative-analysis with user-friendly features that are rooted in best practices from multivariate statistics. Results We developed a Java based Essential Dynamics toolkit called JED to compare the ED from multiple protein trajectories. Trajectories from different simulations and different proteins can be pooled for comparative studies. JED implements Cartesian-based coordinates (cPCA) and internal distance pair coordinates (dpPCA) as options to construct covariance (Q) or correlation (R) matrices. Statistical methods are implemented for treating outliers, benchmarking sampling adequacy, characterizing the precision of Q and R, and reporting partial correlations. JED output results as text files that include transformed coordinates for aligned structures, several metrics that quantify protein mobility, PCA modes with their eigenvalues, and displacement vector (DV) projections onto the top principal modes. Pymol scripts together with PDB files allow movies of individual Q- and R-cPCA modes to be visualized, and the essential dynamics occurring within user-selected time scales. Subspaces defined by the top eigenvectors are compared using several statistical metrics to quantify similarity/overlap of high dimensional vector spaces. Free energy landscapes can be generated for both cPCA and dpPCA. Conclusions JED offers a convenient toolkit that encourages best practices in applying multivariate statistics methods to perform comparative studies of essential dynamics over multiple proteins. For each protein, Cartesian coordinates or internal distance pairs can be employed over the entire structure or user-selected parts to quantify similarity/differences in mobility and correlations in dynamics to develop insight into protein structure/function relationships.

Low-rank approximations to a real matrix A can be computed from ZZT A, where Z is a matrix with orthonormal columns, and the accuracy of the approximation can be estimated from some norm of A - ZZT A. We show that computing A - ZZT A in the two-norm, Frobenius norms, and more generally any Schatten p-norm is a well-posed mathematical problem; and, in contrast to dominant subspace computations, it does not require a singular value gap. We also show that this problem is well-conditioned (insensitive) to additive perturbations in A and Z, and to dimension-changing or multiplicative perturbations in A--regardless of the accuracy of the approximation. For the special case when A does indeed have a singular values gap, connections are established between low-rank approximations and subspace angles. [ABSTRACT FROM AUTHOR]

A set of lines through the origin is called equiangular if every pair of lines defines the same angle, and the maximum size of an equiangular set of lines in Rn was studied extensively for the last 70 years. In this paper, we study analogous questions for k-dimensional subspaces. We discuss natural ways of defining the angle between k-dimensional subspaces and correspondingly study the maximum size of an equiangular set of k-dimensional subspaces in Rn. Our bounds extend and improve a result of Blokhuis. [ABSTRACT FROM AUTHOR]

We study the classification performance of Kronecker-structured (K-S) subpsace models in two asymptotic regimes and develop an algorithm for fast and compact K-S subspace learning for better classification and representation of multidimensional signals by exploiting the structure in the signal. First, we study the classification performance in terms of diversity order and pairwise geometry of the subspaces. We derive an exact expression for the diversity order as a function of the signal and subspace dimensions of a K-S model. Next, we study the classification capacity, the maximum rate at which the number of classes can grow as the signal dimension goes to infinity. Then, we describe a fast algorithm for Kronecker-structured learning of discriminative dictionaries (K-SLD $^2$). Finally, we evaluate the empirical classification performance of K-S models for the synthetic data, showing that they agree with the diversity order analysis. We also evaluate the performance of K-SLD $^2$ on synthetic and real-world datasets showing that the K-SLD $^2$ balances compact signal representation and good classification performance. [ABSTRACT FROM AUTHOR]

Given a real symmetric positive semidefinite matrix E, and an approximation S that is a sum of n independent matrix-valued random variables, we present bounds on the relative error in S due to randomization. The bounds do not depend on the matrix dimensions but only on the numerical rank (intrinsic dimension) of E. Our approach resembles the low-rank approximation of kernel matrices from random features, but our accuracy measures are more stringent. In the context of parameter selection based on active subspaces, where S is computed via Monte Carlo sampling, we present a bound on the number of samples so that with high probability the angle between the dominant subspaces of E and S is less than a user-specified tolerance. This is a substantial improvement over existing work, as it is a nonasymptotic and fully explicit bound on the sampling amount n, and it allows the user to tune the success probability. It also suggests that Monte Carlo sampling can be efficient in the presence of many parameters, as long as the underlying function f is sufficiently smooth. [ABSTRACT FROM AUTHOR]

This paper is concerned with approximating the dominant left singular vector space of a real matrix A of arbitrary dimension, from block Krylov spaces generated by the matrix AAT and the block vector AX. Two classes of results are presented. First are bounds on the distance, in the twoand Frobenius norms, between the Krylov space and the target space. The distance is expressed in terms of principal angles. Second are bounds for the low-rank approximation computed from the Krylov space compared to the best low-rank approximation, in the twoand Frobenius norms. For starting guesses X of full column-rank, the bounds depend on the tangent of the principal angles between X and the dominant right singular vector space of A. The results presented here form the structural foundation for the analysis of randomized Krylov space methods. The innovative feature is a combination of traditional Lanczos convergence analysis with optimal approximations via least squares problems. [ABSTRACT FROM AUTHOR]

Dimension reduction plays an essential role when decreasing the complexity of solving large-scale problems. The well-known Johnson-Lindenstrauss (JL) lemma and restricted isometry property (RIP) admit the use of random projection to reduce the dimension while keeping the Euclidean distance, which leads to the boom of compressed sensing and the field of sparsity related signal processing. Recently, successful applications of sparse models in computer vision and machine learning have increasingly hinted that the underlying structure of high dimensional data looks more like a union of subspaces. In this paper, motivated by JL lemma and an emerging field of compressed subspace clustering, we study for the first time the RIP of Gaussian random matrices for the compression of two subspaces based on the generalized projection F -norm distance. We theoretically prove that with high probability the affinity or distance between two projected subspaces are concentrated around their estimates. When the ambient dimension after projection is sufficiently large, the affinity and distance between two subspaces almost remain unchanged after random projection. Numerical experiments verify the theoretical work. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space of dimension not less than 3 and let G k ( H ) be the Grassmannian formed by k -dimensional subspaces of H . Suppose that dim ⁡ H ≥ 2 k > 2 . We show that the transformations of G k ( H ) induced by linear or conjugate-linear isometries can be characterized as transformations preserving some of principal angles (corresponding to the orthogonality, adjacency and ortho-adjacency relations). As a consequence, we get the following: if the dimension of H is finite and greater than 2 k , then every transformation of G k ( H ) preserving the orthogonality relation in both directions is a bijection induced by a unitary or anti-unitary operator. [ABSTRACT FROM AUTHOR]

One of the most efficient interior-point methods for some classes of block-angular structured problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. In this work we show that the choice of a good preconditioner depends on geometrical properties of the constraint structure. In particular, the principal angles between the subspaces generated by the diagonal blocks and the linking constraints can be used to estimate ex ante the efficiency of the preconditioner. Numerical validation is provided with some generated optimization problems. An application to the solution of multicommodity network flow problems with nodal capacities and equal flows of up to 64 million variables and up to 7.9 million constraints is also presented. These computational results also show that predictor-corrector directions combined with iterative system solves can be a competitive option for large instances. [ABSTRACT FROM AUTHOR]

Background: Essential Dynamics (ED) is a common application of principal component analysis (PCA) to extract biologically relevant motions from atomic trajectories of proteins. Covariance and correlation based PCA are two common approaches to determine PCA modes (eigenvectors) and their eigenvalues. Protein dynamics can be characterized in terms of Cartesian coordinates or internal distance pairs. In understanding protein dynamics, a comparison of trajectories taken from a set of proteins for similarity assessment provides insight into conserved mechanisms. Comprehensive software is needed to facilitate comparative-analysis with user-friendly features that are rooted in best practices from multivariate statistics. Results: We developed a Java based Essential Dynamics toolkit called JED to compare the ED from multiple protein trajectories. Trajectories from different simulations and different proteins can be pooled for comparative studies. JED implements Cartesian-based coordinates (cPCA) and internal distance pair coordinates (dpPCA) as options to construct covariance (Q) or correlation (R) matrices. Statistical methods are implemented for treating outliers, benchmarking sampling adequacy, characterizing the precision of Q and R, and reporting partial correlations. JED output results as text files that include transformed coordinates for aligned structures, several metrics that quantify protein mobility, PCA modes with their eigenvalues, and displacement vector (DV) projections onto the top principal modes. Pymol scripts together with PDB files allow movies of individual Q- and R-cPCA modes to be visualized, and the essential dynamics occurring within user-selected time scales. Subspaces defined by the top eigenvectors are compared using several statistical metrics to quantify similarity/overlap of high dimensional vector spaces. Free energy landscapes can be generated for both cPCA and dpPCA. Conclusions: JED offers a convenient toolkit that encourages best practices in applying multivariate statistics methods to perform comparative studies of essential dynamics over multiple proteins. For each protein, Cartesian coordinates or internal distance pairs can be employed over the entire structure or user-selected parts to quantify similarity/differences in mobility and correlations in dynamics to develop insight into protein structure/function relationships. [ABSTRACT FROM AUTHOR]

The natural action of the unitary group U(n) on ℂn induces its action on the Grassmann manifold GkR(ℂn) consisting of real k-dimensional subspaces in ℂn. In [9] it has been shown that the Kähler angle, used by Chern and Wolfson in the theory of minimal surfaces, determines the orbit of a 2-plane of a complex vector space in the real Grassmannian under the action of the unitary group. Generalizing such notion in [15], the multiple Kähler angle θ(U) of a real subspace U of a complex vector space is defined and it is shown that it is a complete invariant with respect to the natural action of the unitary group, that is, for two real subspaces V and W of same dimension in ℂn, there exists g in U (n) which satisfies W = g · V if and only if θ (V) = θθ(W). In this article we determine a complete invariant for a real subspace of dimension 2 with respect to the action of Sp(n) and Sp(n) · Sp(1) - the groups of automorphisms of a real vector space endowed respectively with an Hermitian hypercomplex and an Hermitian quaternionic structure. A model of such spaces is the n-dimensional quaternionic numerical space Hn, a vector space of dimension 4n over ℝ. We introduce the imaginary measure and the characteristic deviation of a 2-plane and prove that they characterize completely the orbit in a 2-plane in GR(ℍn) under the action of the groups Sp(n) and Sp(n) · Sp(1), respectively. [ABSTRACT FROM AUTHOR]

The leverage scores of a full-column rank matrix A are the squared row norms of any orthonormal basis for range (A). We show that corresponding leverage scores of two matrices A and A + ΔA are close in the relative sense if they have large magnitude and if all principal angles between the column spaces of A and A + ΔA are small. We also show three classes of bounds that are based on perturbation results of QR decompositions. They demonstrate that relative differences between individual leverage scores strongly depend on the particular type of perturbation ΔA. The bounds imply that the relative accuracy of an individual leverage score depends on its magnitude and the two-norm condition of A if ΔA is a general perturbation; the two-norm condition number of A if ΔA is a perturbation with the same normwise row-scaling as A; (to first order) neither condition number nor leverage score magnitude if ΔA is a componentwise row-scaled perturbation. Numerical experiments confirm the qualitative and quantitative accuracy of our bounds. [ABSTRACT FROM AUTHOR]

We present an original framework for uncertainty quantification (UQ) in optimization. It is based on a cascade of ingredients with growing computational complexity for both forward and reverse uncertainty propagation. The approach is merely geometric. It starts with a complexity-based splitting of the independent variables and the definition of a parametric optimization problem. Geometric characterization of global sensitivity spaces through their dimensions and relative positions by the principal angles between global search subspaces bring a first set of information on the impact of uncertainties on the functioning parameters on the optimal solution. Joining the multi-point descent direction and the quantiles on the optimization parameters permits to define the notion of Directional Extreme Scenarios (DES) without sampling of large dimension design spaces. One goes beyond DES with Ensemble Kalman Filters (EnKF) after the multi-point optimization algorithm is cast into an ensemble simulation environment. This formulation accounts for the variability in large dimension. The UQ cascade ends with the joint application of the EnKF and DES leading to the concept of Ensemble Directional Extreme Scenarios (EDES) which provides more exhaustive possible extreme scenarios knowing the Probability Density Function of our optimization parameters. A final interest of the approach is that it provides an indication of the size of the ensemble which must be considered in the EnKF. These ingredients are illustrated on an history matching problem. [ABSTRACT FROM AUTHOR]

Let [ABSTRACT FROM AUTHOR]

We study the eigenvalues and eigenspaces of the quadratic matrix polynomial Mλ²+ sDλ + K as s → ∞, where M and K are symmetric positive definite and D is symmetric positive semidefinite. This work is motivated by its application to modal analysis of finite element models with strong linear damping. Our results yield a mathematical explanation of why too strong damping may lead to practically undamped modes such that all nodes in the model vibrate essentially in phase. [ABSTRACT FROM AUTHOR]

We present a fast algorithm for approximate canonical correlation analysis (CCA). Given a pair of tall-and-thin matrices, the proposed algorithm first employs a randomized dimensionality reduction transform to reduce the size of the input matrices, and then applies any CCA algorithm to the new pair of matrices. The algorithm computes an approximate CCA to the original pair of matrices with provable guarantees while requiring asymptotically fewer operations than the state-of-the-art exact algorithms. [ABSTRACT FROM AUTHOR]

This article introduces the canonical decomposition of the vector space of multivariate polynomials for a given monomial ordering. Its importance lies in solving multivariate polynomial systems, computing Gröbner bases, and solving the ideal membership problem. An SVD-based algorithm is presented that numerically computes the canonical decomposition. It is then shown how, by introducing the notion of divisibility into this algorithm, a numerical Gröbner basis can also be computed. In addition, we demonstrate how the canonical decomposition can be used to decide whether the affine solution set of a multivariate polynomial system is zero-dimensional and to solve the ideal membership problem numerically. The SVD-based canonical decomposition algorithm is also extended to numerically compute border bases. A tolerance for each of the algorithms is derived using perturbation theory of principal angles. This derivation shows that the condition number of computing the canonical decomposition and numerical Gröbner basis is essentially the condition number of the Macaulay matrix. Numerical experiments with both exact and noisy coefficients are presented and discussed. [ABSTRACT FROM AUTHOR]

The paper considers robust parametric optimization problems using multi-point formulations and addresses the issue of the approximation of the gradient of the functional by reduced order models. The question of interest is the impact of such approximations on the search subspace in the multi-point optimization problem. The mathematical concept used to evaluate these approximations is the principal angles between subspaces and practical ways to evaluate these are provided. An additional indicator is provided when a descent minimization algorithm is used. The approach appears also to be an interesting tool for uncertainty quantification of the design in the presence of models of increasing complexity. The application of these concepts is illustrated in the design of the shape of an aircraft robust over a range of transverse winds. [ABSTRACT FROM AUTHOR]

We extend Jordan's notion of principal angles to work for two subspaces of quaternionic space, and so have a method to analyze two orthogonal projections in the matrices over the real, complex or quaternionic field (or skew field). From this we derive an algorithm to turn almost commuting projections into commuting projections that minimizes the sum of the displacements of the two projections. We quickly prove what we need using the universal real C*-algebra generated by two projections. [ABSTRACT FROM AUTHOR]