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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.
|
[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, (1980). Google Scholar |
[4] |
L. Elsner, On some algebraic problems in connection with general eigenvalue algorithms,, Lin. Alg. Appl., 26 (1979), 123.
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.
doi: 10.1016/0024-3795(82)90109-4. |
[6] |
I. Gohberg, S. Goldberg and M. A. Kaashoek, "Basic Classes of Linear Operators,", Birkhäuser Verlag, (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, 141 (2003), 1.
|
[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.
|
[9] |
M. Lindner, "Infinite Matrices and Their Finite Sections. An Introduction to the Limit Operator Method,", Frontiers in Mathematics, (2006).
|
[10] |
M. Lindner and G. Strang, The main diagonal of a permutation matrix,, Linear Algebra and Its Applications, 439 (2013), 524.
doi: 10.1016/j.laa.2012.02.034. |
[11] |
G. Panova, Factorization of banded permutations,, Proceedings Amer. Math. Soc., 140 (2012), 3805.
doi: 10.1090/S0002-9939-2012-11411-X. |
[12] |
J. Plemelj, Riemannsche Funktionenscharen mit gegebener Monodromiegruppe,, Monat. Math. Phys., 19 (1908), 211.
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.
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 (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.
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, 51 (2012), 771.
|
[18] |
G. Strang, Fast transforms: Banded matrices with banded inverses,, Proc. Natl. Acad. Sci., 107 (2010), 12413.
doi: 10.1073/pnas.1005493107. |
[19] |
G. Strang, Groups of banded matrices with banded inverses,, Proceedings Amer. Math. Soc., 139 (2011), 4255.
doi: 10.1090/S0002-9939-2011-10959-6. |
[20] |
G. Strang, The algebra of elimination,, manuscript, (2012). Google Scholar |
[21] |
G. Strang and Tri-Dung Nguyen, The interplay of ranks of submatrices,, SIAM Review, 46 (2004), 637.
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.
|
[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, (1980). Google Scholar |
[4] |
L. Elsner, On some algebraic problems in connection with general eigenvalue algorithms,, Lin. Alg. Appl., 26 (1979), 123.
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.
doi: 10.1016/0024-3795(82)90109-4. |
[6] |
I. Gohberg, S. Goldberg and M. A. Kaashoek, "Basic Classes of Linear Operators,", Birkhäuser Verlag, (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, 141 (2003), 1.
|
[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.
|
[9] |
M. Lindner, "Infinite Matrices and Their Finite Sections. An Introduction to the Limit Operator Method,", Frontiers in Mathematics, (2006).
|
[10] |
M. Lindner and G. Strang, The main diagonal of a permutation matrix,, Linear Algebra and Its Applications, 439 (2013), 524.
doi: 10.1016/j.laa.2012.02.034. |
[11] |
G. Panova, Factorization of banded permutations,, Proceedings Amer. Math. Soc., 140 (2012), 3805.
doi: 10.1090/S0002-9939-2012-11411-X. |
[12] |
J. Plemelj, Riemannsche Funktionenscharen mit gegebener Monodromiegruppe,, Monat. Math. Phys., 19 (1908), 211.
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.
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 (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.
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, 51 (2012), 771.
|
[18] |
G. Strang, Fast transforms: Banded matrices with banded inverses,, Proc. Natl. Acad. Sci., 107 (2010), 12413.
doi: 10.1073/pnas.1005493107. |
[19] |
G. Strang, Groups of banded matrices with banded inverses,, Proceedings Amer. Math. Soc., 139 (2011), 4255.
doi: 10.1090/S0002-9939-2011-10959-6. |
[20] |
G. Strang, The algebra of elimination,, manuscript, (2012). Google Scholar |
[21] |
G. Strang and Tri-Dung Nguyen, The interplay of ranks of submatrices,, SIAM Review, 46 (2004), 637.
doi: 10.1137/S0036144503434381. |
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