Results 1 
9 of
9
The many proofs of an identity on the norm of oblique projections
 Numer. Algorithms
"... Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P 2 = P, which is neither null nor the identity, it holds that ‖P ‖ = ‖I − P ‖. This useful equality, while not widelyknown, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler o ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
(Show Context)
Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P 2 = P, which is neither null nor the identity, it holds that ‖P ‖ = ‖I − P ‖. This useful equality, while not widelyknown, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler ones are presented.
PRINCIPAL ANGLES AND APPROXIMATION 177
"... This paper is dedicated to Professor Tsuyoshi Ando, in celebration of his expertise in matrix and operator theory Communicated by G. Androulakis Abstract. 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 pr ..."
Abstract
 Add to MetaCart
(Show Context)
This paper is dedicated to Professor Tsuyoshi Ando, in celebration of his expertise in matrix and operator theory Communicated by G. Androulakis Abstract. 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. 1. Two projections, the threefold way The general form of two projections on complex Hilbert space is wellknown, going back to at least Dixmier [6]. The real case is older, being implicit in the work of Jordan [13, §IV], where principal vectors and principal angles were introduced. From principal vectors one can derive the structure theorem for matrix projections, as is explained in the real case in [8]. Restricted to the finitedimensional case, we can think of these as theorems about two projections in certain finitedimensional real C∗algebras. One would therefore expect the same result to hold in all finitedimensional real C∗algebras, and so in Mn(H) where H is the skew field of quaternions.
A NEW DECOMPOSITION FOR SQUARE MATRICES
, 2010
"... A new decomposition is derived for any complex square matrix. This decomposition is based on the canonical angles between the column space of this matrix and the column space of its conjugate transpose. Some applications of this factorization are given; in particular some matrix partial orderings ..."
Abstract
 Add to MetaCart
(Show Context)
A new decomposition is derived for any complex square matrix. This decomposition is based on the canonical angles between the column space of this matrix and the column space of its conjugate transpose. Some applications of this factorization are given; in particular some matrix partial orderings and the relationship between the canonical angles and various classes of matrices are studied.
On the Strengthening of Topological Signals in Persistent Homology through Vector Bundle Based Maps
"... Persistent homology is a relatively new tool from topological data analysis that has transformed, for many, the way data sets (and the information contained in those sets) are viewed. It is derived directly from techniques in computational homology but has the added feature that it is able to captu ..."
Abstract
 Add to MetaCart
Persistent homology is a relatively new tool from topological data analysis that has transformed, for many, the way data sets (and the information contained in those sets) are viewed. It is derived directly from techniques in computational homology but has the added feature that it is able to capture structure at multiple scales. One way that this multiscale information can be presented is through a barcode. A barcode consists of a collection of line segments each representing the range of parameter values over which a generator of a homology group persists. A segment’s length relative to the lenght of other segments is an indication of the strength of a corresponding topological signal. In this paper, we consider how vector bundles may be used to reembed data as a means to improve the topological signal. As an illustrative example, we construct maps of tori to a sequence of Grassmannians of increasing dimension. We equip the Grassmannian with the geodesic metric and observe an improvement in barcode signal strength as the dimension of the Grassmannians increase. 2
unknown title
, 2006
"... The many proofs of an identity on the norm of oblique projections ..."
(Show Context)
ELA A NEW DECOMPOSITION FOR SQUARE MATRICES∗
"... Abstract. A new decomposition is derived for any complex square matrix. This decomposition is based on the canonical angles between the column space of this matrix and the column space of its conjugate transpose. Some applications of this factorization are given; in particular some matrix partial or ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. A new decomposition is derived for any complex square matrix. This decomposition is based on the canonical angles between the column space of this matrix and the column space of its conjugate transpose. Some applications of this factorization are given; in particular some matrix partial orderings and the relationship between the canonical angles and various classes of matrices are studied. Key words. Decomposition of matrices, EP matrices, Canonical angles, Matrix partial ordering. AMS subject classifications. 15A23, 15A57. 1. Introduction. Let Cm,n be the set of m × n complex matrices, and let A∗, R(A), N (A), and rank(A) denote the conjugate transpose, column space, null space, and rank, respectively, of A ∈ Cm,n. For a nonsingular A ∈ Cn,n, we shall denote A− ∗ = (A−1) ∗ = (A∗)−1. Furthermore, let A † stand for the MoorePenrose inverse