Article

# Determinants of block tridiagonal matrices

Dipartimento di Fisica, Università degli Studi di Milano and INFN, Sezione di Milano, Via Celoria 16, Milano, Italy

Linear Algebra and its Applications (Impact Factor: 0.98). 01/2008; DOI: 10.1016/j.laa.2008.06.015 Source: arXiv

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**ABSTRACT:**The blackout game(Lightout Game, Merlin Game, σ+Game) is a popular game on a squareboard. When we toggle a button with black or white color, it changes the color of itself and other buttons which have common edges. It is similar to the "Reversi(Othello) Game". With this rule, we can win the game when we have a squareboard with all same colors after some clicks. Here we show that the winning conditions for the general m × n blackout games are related with the determinant of a block triangular matrix generated by a given blackout game. The Fibonacci sequences are used to get the determinant of the block trian-gular matrix. We investigate some properties of a generalized Fibonacci sequences with a winnable condition for the blackout game. Also, we introduce a JAVA simulation tool that gives us winnable conditions for an arbitrary given m × n blackout game.International Journal of Contemporary Mathematical Sciences. 01/2010; 5(29). - [Show abstract] [Hide abstract]

**ABSTRACT:**The formulas presented in [Molinari, L.G. Determinants of block tridiagonal matri-ces. Linear Algebra Appl., 2008; 429, 2221–2226] for evaluating the determinant of block tridiagonal matrices with (or without) corners are used to derive the determinant of any multidiagonal matri-ces with (or without) corners with some specified non-zero minors. Algorithms for calculation the determinant based on this method are given and properties of the determinants are studied. Some applications are presented.The electronic journal of linear algebra ELA 11/2012; 25:101-117. · 0.89 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A Krawtchouk polynomial is introduced as the classical Mac-Williams identity, which can be expressed in weight-enumerator-free form of a linear code and its dual code over a Hamming scheme. In this paper we find a new explicit expression for the -number and the -number, which are more generalized notions of the Krawtchouk polynomial in the P-polynomial schemes by using an extended version of a discrete Green's function. As corollaries, we obtain a new expression of the Krawtchouk polynomial over the Hamming scheme and the Eberlein polynomial over the Johnson scheme. Furthermore, we find another version of the MacWilliams identity over a Hamming scheme.Journal of the Korean Mathematical Society 05/2013; 50(3). · 0.42 Impact Factor

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