The main diagonal of a permutation matrix. (English) Zbl 1283.15092
Summary: By counting 1’s in the “right half” of \(2w\) consecutive rows, we locate the main diagonal of any doubly infinite permutation matrix with bandwidth \(w\). Then the matrix can be correctly centered and factored into block-diagonal permutation matrices. {
} Part II of the paper discusses the same questions for the much larger class of band-dominated matrices. The main diagonal is determined by the Fredholm index of a singly infinite submatrix. Thus the main diagonal is determined “at infinity” in general, but from only \(2w\) rows for banded permutations.
MSC:
| 15B34 | Boolean and Hadamard matrices |
Keywords:
banded matrix; infinite matrix; main diagonal; factorization; permutation matrix; band-dominated matrices; Fredholm indexReferences:
| [1] | Albert, C.; Li, Chi-Kwong; Strang, G.; Yu, Gexin, Permutations as products of parallel transpositions, SIAM J. Discrete Math., 25, 1412-1417 (2011) · Zbl 1237.05002 |
| [2] | Asplund, E., Inverses of matrices \(\{a_{ij} \}\) which satisfy \(a_{ij} = 0\) for \(j > i + p\), Math. Scand., 7, 57-60 (1959) · Zbl 0093.24102 |
| [3] | Böttcher, A., Infinite matrices and projection methods, (Lancaster, P., Lectures on Operator Theory and its Applications. Lectures on Operator Theory and its Applications, Fields Institute Monographs, vol. 3 (1996), Amer. Math. Soc.), 1-72 · Zbl 0842.65032 |
| [4] | Böttcher, A.; Grudsky, S. M., Spectral Properties of Banded Toeplitz Matrices (2005), SIAM · Zbl 1089.47001 |
| [5] | Böttcher, A.; Silbermann, B., Introduction to Large Truncated Toeplitz Matrices (1999), Springer · Zbl 0916.15012 |
| [6] | Chandler-Wilde, S. N.; Lindner, M., Limit operators, collective compactness, and the spectral theory of infinite matrices, Mem. Amer. Math. Soc., 210, 989 (2011) · Zbl 1219.47001 |
| [7] | Davies, E. B., Linear Operators and Their Spectra (2007), Cambridge University Press · Zbl 1138.47001 |
| [8] | de Boor, C., What is the main diagonal of a biinfinite band matrix?, (DeVore, R.; Scherer, K., Quantitative Approximation (1980), Academic Press), 11-23 · Zbl 0457.41014 |
| [9] | Gohberg, I.; Feldman, I. A., Convolution equations and projection methods for their solution, Transl. of Math. Monographs, vol. 41 (1974), Amer. Math. Soc., [Russian original: Nauka, Moscow, 1971] · Zbl 0278.45008 |
| [10] | Gohberg, I.; Goldberg, S.; Kaashoek, M. A., Classes of Linear Operators, vol. 1of 2 (1993), Birkhäuser · Zbl 0789.47001 |
| [11] | Heinig, G.; Hellinger, F., The finite section method for Moore-Penrose inversion of Toeplitz operators, Integral Equations Operator Theory, 19, 419-446 (1994) · Zbl 0817.47036 |
| [12] | Kurbatov, V. G., Functional Differential Operators and Equations (1999), Kluwer · Zbl 0926.34053 |
| [13] | Lindner, M., Infinite Matrices and Their Finite Sections: An Introduction to the Limit Operator Method. Infinite Matrices and Their Finite Sections: An Introduction to the Limit Operator Method, Frontiers in Mathematics (2006), Birkhäuser · Zbl 1107.47001 |
| [14] | Lindner, M., Fredholmness and index of operators in the Wiener algebra are independent of the underlying space, Oper. Matrices, 2, 297-306 (2008) · Zbl 1153.47008 |
| [15] | Lindner, M., The finite section method and stable subsequences, Appl. Numer. Math., 60, 501-512 (2010) · Zbl 1197.47029 |
| [17] | Rabinovich, V. S.; Roch, S.; Roe, J., Fredholm indices of band-dominated operators, Integral Equations Operator Theory, 49, 221-238 (2004) · Zbl 1068.47016 |
| [18] | Rabinovich, V. S.; Roch, S.; Silbermann, B., Limit Operators and Their Applications in Operator Theory (2004), Birkhäuser · Zbl 1077.47002 |
| [19] | Roch, S., Finite sections of band-dominated operators, Mem. Amer. Math. Soc., 191, 895 (2008) · Zbl 1167.47015 |
| [21] | Strang, G., Fast transforms: banded matrices with banded inverses, Proc. Natl. Acad. Sci., 107, 28, 12413-12416 (2010) · Zbl 1205.65348 |
| [22] | Strang, G., Groups of banded matrices with banded inverses, Proc. Amer. Math. Soc., 139, 4255-4264 (2011) · Zbl 1241.15011 |
| [25] | Strang, G.; Nguyen, Tri, The interplay of ranks of submatrices, SIAM Rev., 46, 637-646 (2004) · Zbl 1063.15002 |
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