"Matrix Mathematics" is a reference work for users of matrices in all branches of engineering, science, and applied mathematics. This book brings together a vast body of results on matrix theory for easy reference and immediate application. Each chapter begins with the development of relevant background theory followed by a large collection of specialized results. Hundreds of identities, inequalities, and matrix facts are stated rigorously and clearly with cross references, citations to the literature, and illuminating remarks. Twelve chapters cover all of the major topics in matrix theory: preliminaries; basic matrix properties; matrix classes and transformations; matrix polynomials and rational transfer functions; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; norms; functions of matrices and their derivatives; the matrix exponential and stability theory; and linear systems and control theory. A detailed list of symbols, a summary of notation and conventions, an extensive bibliography with author index, and an extensive index are provided for ease of use. The book will be useful for students at both the undergraduate and graduate levels, as well as for researchers and practitioners in all branches of engineering, science, and applied mathematics.

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Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory

ESPRIT is a high-resolution signal parameter estimation technique based on the translational invariance structure of a sensor array. Previous ESPRIT algorithms do not use the fact that the operator representing the phase delays between the two subarrays is unitary. The authors present a simple and efficient method to constrain the estimated phase factors to the unit circle, if centro-symmetric array configurations are used. Unitary ESPRIT, the resulting closed-form algorithm, has an ESPRIT-like structure except for the fact that it is formulated in terms of real-valued computations throughout. Since the dimension of the matrices is not increased, this completely real-valued algorithm achieves a substantial reduction of the computational complexity. Furthermore, Unitary ESPRIT incorporates forward-backward averaging, leading to an improved performance compared to the standard ESPRIT algorithm, especially for correlated source signals. Like standard ESPRIT, Unitary ESPRIT offers an inexpensive possibility to reconstruct the impinging wavefronts (signal copy). These signal estimates are more accurate, since Unitary ESPRIT improves the underlying signal subspace estimates. Simulations confirm that, even for uncorrelated signals, the standard ESPRIT algorithm needs twice the number of snapshots to achieve a precision comparable to that of Unitary ESPRIT. Thus, Unitary ESPRIT provides increased estimation accuracy with a reduced computational burden. >

Unitary ESPRIT: how to obtain increased estimation accuracy with a reduced computational burden

Multidimensional harmonic retrieval problems are encountered in a variety of signal processing applications including radar, sonar, communications, medical imaging, and the estimation of the parameters of the dominant multipath components from MIMO channel measurements. R-dimensional subspace-based methods, such as R-D Unitary ESPRIT, R-D RARE, or R-D MUSIC, are frequently used for this task. Since the measurement data is multidimensional, current approaches require stacking the dimensions into one highly structured matrix. However, in the conventional subspace estimation step, e.g., via an SVD of the latter matrix, this structure is not exploited. In this paper, we define a measurement tensor and estimate the signal subspace through a higher-order SVD. This allows us to exploit the structure inherent in the measurement data already in the first step of the algorithm which leads to better estimates of the signal subspace. We show how the concepts of forward-backward averaging and the mapping of centro-Hermitian matrices to real-valued matrices of the same size can be extended to tensors. As examples, we develop the R-D standard Tensor-ESPRIT and the R-D Unitary Tensor-ESPRIT algorithms. However, these new concepts can be applied to any multidimensional subspace-based parameter estimation scheme. Significant improvements of the resulting parameter estimation accuracy are achieved if there is at least one of the R dimensions, which possesses a number of sensors that is larger than the number of sources. This can already be observed in the two-dimensional case.

Higher-Order SVD-Based Subspace Estimation to Improve the Parameter Estimation Accuracy in Multidimensional Harmonic Retrieval Problems

In this paper, the authors give explicit inverse formulae for 2 × 2 block matrices with three different partitions. Then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as Hamiltonian, per-Hermitian, and centro-Hermitian matrices.

Inverses of 2 × 2 block matrices

This paper presents a new Jacobi-type method to calculate a simultaneous Schur decomposition (SSD) of several real-valued, nonsymmetric matrices by minimizing an appropriate cost function Thereby, the SSD reveals the "average eigenstructure" of these nonsymmetric matrices This enables an R-dimensional extension of Unitary ESPRIT to estimate several undamped R-dimensional modes or frequencies along with their correct pairing in multidimensional harmonic retrieval problems Unitary ESPRIT is an ESPRIT-type high-resolution frequency estimation technique that is formulated in terms of real-valued computations throughout For each of the R dimensions, the corresponding frequency estimates are obtained from the real eigenvalues of a real-valued matrix The SSD jointly estimates the eigenvalues of all R matrices and, thereby, achieves automatic pairing of the estimated R-dimensional modes via a closed-form procedure that neither requires any search nor any other heuristic pairing strategy Moreover, we describe how R-dimensional harmonic retrieval problems (with R/spl ges/3) occur in array signal processing and model-based object recognition applications

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Simultaneous Schur decomposition of several nonsymmetric matrices to achieve automatic pairing in multidimensional harmonic retrieval problems

Centrohermitian and skew-centrohermitian matrices

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Note on the spectra of finite permutation matrices

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Transformationsinvariante Matrizen

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On regular polynomial matrices. I. A new criterion for a polynomial matrix to be simple

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Anna Lee

Papers

Centrohermitian and skew-centrohermitian matrices

Note on the spectra of finite permutation matrices

Transformationsinvariante Matrizen

On regular polynomial matrices. I. A new criterion for a polynomial matrix to be simple

On regular polynomial matrices. II. The associated matrix pencils; simple matrix pencils