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Three steps on an open road
| 1. | Massachusetts Institute of Technology, Cambridge, MA 02139,, United States |
1. If $A$ has a banded inverse;: $A = BC$ with block--diagonal factors $B$ and $C$.
2. Permutations factor into a shift times $N < 2w$ tridiagonal permutations.
3. $A = LPU$ with lower triangular $L$, permutation $P$, upper triangular $U$.
We include examples and references and outlines of proofs.
References:
| [1] |
E. Asplund, Inverses of matrices {$a_{ij}$} which satisfy $a_{ij}=0$ for $j > i + p$, Math. Scand., 7 (1959), 57-60. |
| [2] |
S. N. Chandler-Wilde and M. Lindner, Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices, Amer. Math. Society Memoirs, 210 (2011).
doi: 10.1090/S0065-9266-2010-00626-4. |
| [3] |
C. de Boor, What is the main diagonal of a biinfinite band matrix?, in "Quantitative Approximation" (eds. R. A. DeVore and K. Scherer), Academic Press, 1980. |
| [4] |
L. Elsner, On some algebraic problems in connection with general eigenvalue algorithms, Lin. Alg. Appl., 26 (1979), 123-138.
doi: 10.1016/0024-3795(79)90175-7. |
| [5] |
I. Gohberg and S. Goldberg, Finite dimensional Wiener-Hopf equations and factorizations of matrices, Lin. Alg. Appl., 48 (1982), 219-236.
doi: 10.1016/0024-3795(82)90109-4. |
| [6] |
I. Gohberg, S. Goldberg and M. A. Kaashoek, "Basic Classes of Linear Operators," Birkhäuser Verlag, Basel, 2003.
doi: 10.1007/978-3-0348-7980-4. |
| [7] |
I. Gohberg, M. Kaashoek and I. Spitkovsky, An overview of matrix factorization theory and operator application, in "Factorization and Integrable Systems" (Faro, 2000), Operator Th. Adv. Appl., 141, Birkhäuser, Basel, (2003), 1-102. |
| [8] |
L. Yu. Kolotilina and A. Yu. Yeremin, Bruhat decomposition and solution of sparse linear algebra systems, Soviet J. Numer. Anal. Math. Modelling, 2 (1987), 421-436. |
| [9] |
M. Lindner, "Infinite Matrices and Their Finite Sections. An Introduction to the Limit Operator Method," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. |
| [10] |
M. Lindner and G. Strang, The main diagonal of a permutation matrix, Linear Algebra and Its Applications, 439 (2013), 524-537.
doi: 10.1016/j.laa.2012.02.034. |
| [11] |
G. Panova, Factorization of banded permutations, Proceedings Amer. Math. Soc., 140 (2012), 3805-3812.
doi: 10.1090/S0002-9939-2012-11411-X. |
| [12] |
J. Plemelj, Riemannsche Funktionenscharen mit gegebener Monodromiegruppe, Monat. Math. Phys., 19 (1908), 211-245.
doi: 10.1007/BF01736697. |
| [13] |
V. S. Rabinovich, S. Roch and J. Roe, Fredholm indices of band-dominated operators, Integral Eqns. Oper. Th., 49 (2004), 221-238.
doi: 10.1007/s00020-003-1285-1. |
| [14] |
V. S. Rabinovich, S. Roch and B. Silbermann, "Limit Operators and Their Applications in Operator Theory," Operator Theory: Advances and Applications, 150, Birkhäuser Verlag, Basel, 2004.
doi: 10.1007/978-3-0348-7911-8. |
| [15] |
V. S. Rabinovich, S. Roch and B. Silbermann, The finite section approach to the index formula for band-dominated operators, Operator Theory, 187 (2008), 185-193.
doi: 10.1007/978-3-7643-8893-5_11. |
| [16] |
S. Roch, Finite sections of band-dominated operators, AMS Memoirs, 191 (2008).
doi: 10.1090/memo/0895. |
| [17] |
G. Strang, Banded matrices with banded inverses and $A=LPU$, in "Fifth International Congress of Chinese Mathematicians. Part 1, 2," AMS/IP Studies in Advanced Mathematics, 51, American Math. Society, Providence, RI, (2012), 771-784. |
| [18] |
G. Strang, Fast transforms: Banded matrices with banded inverses, Proc. Natl. Acad. Sci., 107 (2010), 12413-12416.
doi: 10.1073/pnas.1005493107. |
| [19] |
G. Strang, Groups of banded matrices with banded inverses, Proceedings Amer. Math. Soc., 139 (2011), 4255-4264.
doi: 10.1090/S0002-9939-2011-10959-6. |
| [20] |
G. Strang, The algebra of elimination, manuscript, 2012. |
| [21] |
G. Strang and Tri-Dung Nguyen, The interplay of ranks of submatrices, SIAM Review, 46 (2004), 637-646.
doi: 10.1137/S0036144503434381. |
show all references
References:
| [1] |
E. Asplund, Inverses of matrices {$a_{ij}$} which satisfy $a_{ij}=0$ for $j > i + p$, Math. Scand., 7 (1959), 57-60. |
| [2] |
S. N. Chandler-Wilde and M. Lindner, Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices, Amer. Math. Society Memoirs, 210 (2011).
doi: 10.1090/S0065-9266-2010-00626-4. |
| [3] |
C. de Boor, What is the main diagonal of a biinfinite band matrix?, in "Quantitative Approximation" (eds. R. A. DeVore and K. Scherer), Academic Press, 1980. |
| [4] |
L. Elsner, On some algebraic problems in connection with general eigenvalue algorithms, Lin. Alg. Appl., 26 (1979), 123-138.
doi: 10.1016/0024-3795(79)90175-7. |
| [5] |
I. Gohberg and S. Goldberg, Finite dimensional Wiener-Hopf equations and factorizations of matrices, Lin. Alg. Appl., 48 (1982), 219-236.
doi: 10.1016/0024-3795(82)90109-4. |
| [6] |
I. Gohberg, S. Goldberg and M. A. Kaashoek, "Basic Classes of Linear Operators," Birkhäuser Verlag, Basel, 2003.
doi: 10.1007/978-3-0348-7980-4. |
| [7] |
I. Gohberg, M. Kaashoek and I. Spitkovsky, An overview of matrix factorization theory and operator application, in "Factorization and Integrable Systems" (Faro, 2000), Operator Th. Adv. Appl., 141, Birkhäuser, Basel, (2003), 1-102. |
| [8] |
L. Yu. Kolotilina and A. Yu. Yeremin, Bruhat decomposition and solution of sparse linear algebra systems, Soviet J. Numer. Anal. Math. Modelling, 2 (1987), 421-436. |
| [9] |
M. Lindner, "Infinite Matrices and Their Finite Sections. An Introduction to the Limit Operator Method," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. |
| [10] |
M. Lindner and G. Strang, The main diagonal of a permutation matrix, Linear Algebra and Its Applications, 439 (2013), 524-537.
doi: 10.1016/j.laa.2012.02.034. |
| [11] |
G. Panova, Factorization of banded permutations, Proceedings Amer. Math. Soc., 140 (2012), 3805-3812.
doi: 10.1090/S0002-9939-2012-11411-X. |
| [12] |
J. Plemelj, Riemannsche Funktionenscharen mit gegebener Monodromiegruppe, Monat. Math. Phys., 19 (1908), 211-245.
doi: 10.1007/BF01736697. |
| [13] |
V. S. Rabinovich, S. Roch and J. Roe, Fredholm indices of band-dominated operators, Integral Eqns. Oper. Th., 49 (2004), 221-238.
doi: 10.1007/s00020-003-1285-1. |
| [14] |
V. S. Rabinovich, S. Roch and B. Silbermann, "Limit Operators and Their Applications in Operator Theory," Operator Theory: Advances and Applications, 150, Birkhäuser Verlag, Basel, 2004.
doi: 10.1007/978-3-0348-7911-8. |
| [15] |
V. S. Rabinovich, S. Roch and B. Silbermann, The finite section approach to the index formula for band-dominated operators, Operator Theory, 187 (2008), 185-193.
doi: 10.1007/978-3-7643-8893-5_11. |
| [16] |
S. Roch, Finite sections of band-dominated operators, AMS Memoirs, 191 (2008).
doi: 10.1090/memo/0895. |
| [17] |
G. Strang, Banded matrices with banded inverses and $A=LPU$, in "Fifth International Congress of Chinese Mathematicians. Part 1, 2," AMS/IP Studies in Advanced Mathematics, 51, American Math. Society, Providence, RI, (2012), 771-784. |
| [18] |
G. Strang, Fast transforms: Banded matrices with banded inverses, Proc. Natl. Acad. Sci., 107 (2010), 12413-12416.
doi: 10.1073/pnas.1005493107. |
| [19] |
G. Strang, Groups of banded matrices with banded inverses, Proceedings Amer. Math. Soc., 139 (2011), 4255-4264.
doi: 10.1090/S0002-9939-2011-10959-6. |
| [20] |
G. Strang, The algebra of elimination, manuscript, 2012. |
| [21] |
G. Strang and Tri-Dung Nguyen, The interplay of ranks of submatrices, SIAM Review, 46 (2004), 637-646.
doi: 10.1137/S0036144503434381. |
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