# An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing
## 1 Introduction  
Matrix theory is widely used in many disciplines of modern engineering, including wireless communications, signal processing, control, biomedical engineering, etc. From a mathematical perspective, matrices are used both as a representation as well as an information processing tool to solve efficiently and reliably practical engineering problems. Since matrices are prevalent in engineering and have enabled remarkable solutions to problems of paramount importance, this paper overviews some of the major applications of matrix theory in wireless communications and signal processing. Our hope is that this article will help in renewing the interest of engineering community in linear algebra, and in particular in the tools offered by matrix theory, and its applications in engineering. Rather than describing details for all the statements and developments, our goal is instead to provide the readers the basic facts and key features of how matrix techniques were applied in some major engineering applications that played a fundamental role during the last two decades in the fields of signal processing and wireless communications.  
The outline of this paper is summarized as follows. Section 2 describes how matrix techniques are exploited in the modern communication systems. Specifically, we discuss the key role played by the concept of singular value decomposition (SVD) in modeling and characterization of the multiple-input multiple-output (MIMO) or multiple antennas based communication systems, and the important role played by unitary matrix transformations such as Discrete Fourier Transform (DFT) in the orthogonal frequency-division multiplexing (OFDM) signaling scheme, which currently has been adopted in many wireless communications standards. Section 3 focuses on several illustrative uses of matrices in signalestimation and detection applications. In particular, the usefulness of matrix decompositions and matrix inversion lemma in enabling online (recursive) estimation of parameters and development of adaptive filters is illustrated. A special attention is also given to the applications of matrices in sensor array signal processing. A more modern estimation method based on large dimensional random matrix theory is also briefly discussed to exemplify the recent trends in the utilization of matrices in signal processing and wireless communications. Section 4 describes the close relationship between matrices and images. Two image compression methods, which rely on matrix transformation and separation operations, are described. In Section 5, a recently proposed data compression approach termed as compressive (compressed) sensing is briefly examined from the perspective of the role played by matrices. Finally, Section 6 concludes the paper by outlining the key role played by matrices in solving some important problems from the fields of signal processing and wireless communications. Several open research problems that require more advanced linear algebra concepts and results are pointed out as well.  
Throughout this paper, we will denote scalars by lowercase italic letters (\(a\)), vectors by bold lowercase italic letters (\(\mathbf{a}\)), and matrices by bold uppercase Roman letters (\(\mathbf{A}\)).