Computing theCS decomposition of a partitioned orthonormal matrix

1982 ◽  
Vol 40 (3) ◽  
pp. 297-306 ◽  
Author(s):  
G. W. Stewart
Keyword(s):  
Orthonormal Matrix

System Identification Based on Output-Only Decomposition and Subspace Appropriation

2019 ◽  
Vol 141 (9) ◽  
Author(s):  
Amirali Sadeqi ◽  
Shapour Moradi ◽  
Kourosh Heidari Shirazi
Keyword(s):  
Subspace Identification ◽  
Identification Methods ◽  
Input Type ◽  
Stochastic Framework ◽  
Innovation Model ◽  
Orthonormal Matrix ◽  
Output Only ◽  
Row Space ◽  
First Time ◽  
Transient And Steady State

Output-only identification methods have been developed on a stochastic framework, but for the first time, a subspace-based approach is proposed without using geometric and statistical tools. This aids the computational efforts to be significantly reduced and the range of input sources to be extended in a much realistic manner for future output-only analyses. The approach encompasses any input type and can properly work for systems excited by inputs with finite periods. It is demonstrated that the row space of the output sequences spanned by column vectors of the decomposed orthonormal matrix is sufficient to reconstruct the observations. The transient and steady-state portions of the output row space, afterward, can be captured to reconstruct an integrated innovation model. The advantages of the algorithm are highlighted through several numerical and experimental examples comparing with the traditional subspace identification algorithms.

Jacobi Operators and Orthonormal Matrix-Valued Polynomials. I

2017 ◽  
Vol 69 (2) ◽  
pp. 269-282
Author(s):  
R. Hatamleh ◽  
V. A. Zolotarev
Keyword(s):  
Jacobi Operators ◽  
Orthonormal Matrix

scholarly journals Parameterization of orthonormal third-order matrices for linear calibration

2009 ◽  
Vol 57 (6) ◽  
pp. 163-168
Author(s):  
Ivo Moll ◽  
Kateřina Myšková
Keyword(s):  
Rotation Matrix ◽  
Linear Calibration ◽  
Third Order ◽  
The Third ◽  
Rotation Matrices ◽  
Unit Matrix ◽  
Special Cases ◽  
Orthonormal Matrix ◽  
Definition Of ◽  
Real Matrices

The paper derives a parametric definition of the set of third-order orthonormal real matrices.The derivation is done in several partial steps. First a generalized unit matrix is introduced as the simplest case of an orthonormal matrix along with some of its properties and, subsequently, the properties of orthonormal matrices are proved that will be needed.The derivation itself of a parametric definition of third-order orthonormal matrices is based on the numbers of zero entries that are theoretically possible. Therefore, it is first proved that a third-order square matrix with the number of non-zero entries different from nine, eight, five, or three cannot be orthonormal.The number of different ways in which the set of third-order orthonormal matrices can be pa­ra­me­te­ri­zed is greater than one. The concepts of a rotation matrix and a flop-enabling rotation matrix are introduced to motivate the parameterization chosen.Given the product of two rotation matrices and one flop-enabling rotation matrix, it is first proved that it is a third-order orthonormal matrix. In the last part of the paper, it is then proved that such a product already includes, as special cases, all the third-order orthonormal matrices. It is thus a parametric definition of all third-order orthonormal matrices.

Stable recovery of sparse signals with coherent tight frames via lp-analysis approach

2016 ◽  
Vol 15 (04) ◽  
pp. 505-520
Author(s):  
Dekai Liu ◽  
Song Li
Keyword(s):  
Compressed Sensing ◽  
Minimization Problem ◽  
Tight Frame ◽  
Complex Case ◽  
Closed Form Expression ◽  
Sparse Signals ◽  
Tight Frames ◽  
Orthonormal Matrix ◽  
Redundant Dictionaries ◽  
True Signal

In this paper, we consider to recover a signal which is sparse in terms of a tight frame from undersampled measurements via [Formula: see text]-minimization problem for [Formula: see text]. In [Compressed sensing with coherent tight frames via [Formula: see text]-minimization for [Formula: see text], Inverse Probl. Imaging 8 (2014) 761–777], Li and Lin proved that when [Formula: see text] there exists a [Formula: see text], depending on [Formula: see text] such that for any [Formula: see text], each solution of the [Formula: see text]-minimization problem can approximate the true signal well. The constant [Formula: see text] is referred to as the [Formula: see text]-RIP constant of order [Formula: see text] which was first introduced by Candès et al. in [Compressed sensing with coherent and redundant dictionaries, Appl. Comput. Harmon. Anal. 31 (2011) 59–73]. The main aim of this paper is to give the closed-form expression of [Formula: see text]. We show that for every [Formula: see text]-RIP constant [Formula: see text], if [Formula: see text] where [Formula: see text] then the [Formula: see text]-minimization problem can reconstruct the true signal approximately well. Our main results also hold for the complex case, i.e. the measurement matrix, the tight frame and the signal are all in the complex domain. It should be noted that the[Formula: see text]-RIP condition is independent of the coherence of the tight frame (see [Compressed sensing with coherent and redundant dictionaries, Appl. Comput. Harmon. Anal. 31 (2011) 59–73]). In particular, when the tight frame reduces to an identity matrix or an orthonormal matrix, the conclusions in our paper coincide with the results appeared in [Stable recovery of sparse signals via [Formula: see text]-minimization, Appl. Comput. Harmon. Anal. 38 (2015) 161–176].

Jacobi Operators and Orthonormal Matrix-Valued Polynomials. II

2017 ◽  
Vol 69 (6) ◽  
pp. 970-985
Author(s):  
R. Hatamleh ◽  
V. A. Zolotarev
Keyword(s):  
Jacobi Operators ◽  
Orthonormal Matrix
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