Abstract. We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidimensional and does not depend on any embedding into a larger ambient space. Furthermore, it has a concrete expression involving principal angles, and is efficiently computable in numerically stable ways. Our results are largely independent of the Grassmann distance -if desired, it may be substituted by any other common distances between subspaces. Our approach depends on a concrete algebraic geometric view of the Grassmannian that parallels the differential geometric perspective that is well-established in applied and computational mathematics.

Abstract. We show that every n×n matrix is generically a product of ⌊n/2⌋+1 Toeplitz matrices and always a product of at most 2n + 5 Toeplitz matrices. The same result holds true if the word 'Toeplitz' is replaced by 'Hankel', and the generic bound ⌊n/2⌋ + 1 is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not in general replace the subspace of Toeplitz or Hankel matrices by an arbitrary (2n − 1)-dimensional subspace of n × n matrices. Furthermore such decompositions do not exist if we require the factors to be symmetric Toeplitz, persymmetric Hankel, or circulant matrices, even if we allow an infinite number of factors. Lastly, we discuss how the Toeplitz and Hankel decompositions of a generic matrix may be computed by either (i) solving a system of linear and quadratic equations if the number of factors is required to be ⌊n/2⌋ + 1, or (ii) Gaussian elimination in O(n 3 ) time if the number of factors is allowed to be 2n.

We discuss a generalization of the Cohn-Umans method, a potent technique developed for studying the bilinear complexity of matrix multiplication by embedding matrices into an appropriate group algebra. We investigate how the Cohn-Umans method may be used for bilinear operations other than matrix multiplication, with algebras other than group algebras, and we relate it to Strassen's tensor rank approach, the traditional framework for investigating bilinear complexity. To demonstrate the utility of the generalized method, we apply it to find the fastest algorithms for forming structured matrix-vector product, the basic operation underlying iterative algorithms for structured matrices. The structures we study include Toeplitz, Hankel, circulant, symmetric, skew-symmetric, f -circulant, block-Toeplitz-Toeplitz-block, triangular Toeplitz matrices, Toeplitzplus-Hankel, sparse/banded/triangular. Except for the case of skew-symmetric matrices, for which we have only upper bounds, the algorithms derived using the generalized Cohn-Umans method in all other instances are the fastest possible in the sense of having minimum bilinear complexity. We also apply this framework to a few other bilinear operations including matrix-matrix, commutator, simultaneous matrix products, and briefly discuss the relation between tensor nuclear norm and numerical stability.

We study in this article multiplicities of eigenvalues of tensors. There are two natural multiplicities associated to an eigenvalue λ of a tensor: algebraic multiplicity am(λ) and geometric multiplicity gm(λ). The former is the multiplicity of the eigenvalue as a root of the characteristic polynomial, and the latter is the dimension of the eigenvariety (i.e., the set of eigenvectors) corresponding to the eigenvalue.We show that the algebraic multiplicity could change along the orbit of tensors by the orthogonal linear group action, while the geometric multiplicity of the zero eigenvalue is invariant under this action, which is the main difficulty to study their relationships. However, we show that for a generic tensor, every eigenvalue has a unique (up to scaling) eigenvector, and both the algebraic multiplicity and geometric multiplicity are one. In general, we suggest for an m-th order n-dimensional tensor the relationship am(λ) ≥ gm(λ)(m − 1) gm(λ)−1 . We show that it is true for serveral cases, especially when the eigenvariety contains a linear subspace of dimension gm(λ) in coordinate form. As both multiplicities are invariants under the orthogonal linear group action in the matrix counterpart, this generalizes the classical result for a matrix: the algebraic mutliplicity is not smaller than the geometric multiplicity.2010 Mathematics Subject Classification. 15A18; 15A42; 15A69.

The degree of the Grassmannian with respect to the Plücker embedding is well-known. However, the Plücker embedding, while ubiquitous in pure mathematics, is almost never used in applied mathematics. In applied mathematics, the Grassmannian is usually embedded as projection matrices Gr(k, R n ) ∼ = {P ∈ R n×n : P T = P = P 2 , tr(P ) = k} or as involution matrices Gr(k, R n ) ∼ = {X ∈ R n×n : X T = X, X 2 = I, tr(X) = 2k − n}. We will determine an explicit expression for the degree of the Grassmannian with respect to these embeddings. In so doing, we resolved a conjecture of Devriendt-Friedman-Sturmfels about the degree Gr(2, R n ) and in fact generalized it to Gr(k, R n ). We also proved a set theoretic variant of another conjecture of Devriendt-Friedman-Sturmfels about the limit of Gr(k, R n ) in the sense of Gröbner degneration.