In this article we determine the eigenvalues of sequences of tridiagonal matrices that contain a Toeplitz matrix in the upper left block.

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Abstract-This article reviews recent research and current art in the area of auction based mechanisms for electronic procurement. These mechanisms are becoming increasingly relevant in modern day e-procurement systems since they enable a promising way of automating negotiations with suppliers and achieving the ideal goals of procurement efficiency, cost minimization, and agent based deployment. The survey delineates different representative scenarios in e-procurement where auctions can be deployed and describes the conceptual and mathematical aspects of different categories of procurement auctions. We discuss three categories: (1) multi-unit auctions for a single homogeneous type of item; (2) combinatorial procurement auctions where the buyer seeks to procure a bundle of multiple items and the suppliers bid for subsets of the bundle; and (3) multi-attribute auctions where the procurement decisions transcend cost considerations alone, to take into account lead times, logistics costs, and other important attributes. In all three cases, the winner determination problem and the determination of payments turn out to be interesting and challenging combinatorial optimization problems. In our review, we present mathematical formulation of procurement scenarios under each category, bring out the challenge involved in solving the problems, and indicate active research topics. We also present a case study of electronic procurement at General Motors.A Note to Practitioners-Since the burst of the dot.com bubble, many procurement professionals and purchasing managers have begun to question the ability of the Internet to redefine procurement processes within their firms. In this article we set out to show that this would be a misplaced sense of deja vu because the Internet along with a milieu of decision technologies based on Game Theory and Optimization is proving to be a significant tool in the hands of procurement professionals. Sans all the hype, the dot.com phenomena has left behind useful ideas including that of e-platforms for on-line auctions. Building upon this core conceptual construct, familiar to most procurement professionals, we first illustrate the successful implementation of sophisticated auction models by pioneering firms, based on optimization technologies, that meet the requirements of complex business-to-business procurement. We then discuss the exciting field of research this has opened up with a vast potential for immediate and gainful applications. We review the existing stateof-the-art in this field, track its recent developments and classify the models available for different procurement scenarios. We also provide pointers to areas that require further fundamental as well as applied research which calls for the attention of not just academic researchers but also practicing professionals.