The Rook on the Half-Chessboard, or How Not to Diagonalize a Matrix
نویسندگان
چکیده
1/i if i + j ≥ n + 1 0 otherwise, has eigenvalues 1,−1/2, 1/3, . . . , (−1)/n. In Section 2, we establish this result in two ways: (1) by explicitly exhibiting eigenvectors via an odd-looking combinatorial identity, and (2) by giving a matrix that conjugates Tn into an upper triangular matrix. In Section 3, we use the knowledge of the eigenvalues of Tn to study two related random walks, most conveniently summarized as “the rook on the half-chessboard.” Consider an n× n chessboard from which all squares above (but not including) the northwest-southeast diagonal have been removed. In the “sloppy” random walk, a rook is moved as follows. If a coin flip comes up heads, we choose a square uniformly at random in the row containing the rook (including the square the rook occupies), and move the rook there. If the coin flip comes up tails, we do the same using the column instead of the row. A fundamental property of this random walk is the speed at which an initial probability distribution for the position of the rook converges to the uniform distribution. The sloppy walk has an obvious inefficiency built in: two or more moves in the same direction have no more effect on the distribution than a single move in that direction. Hence we also introduce the “ordered” random walk, in which horizontal and vertical moves alternate. We
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