American Institute of Mathematical Sciences

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August  2013, 7(3): 961-966. doi: 10.3934/ipi.2013.7.961

Three steps on an open road

Gilbert Strang 1,

1. 

Massachusetts Institute of Technology, Cambridge, MA 02139,, United States

Received  October 2012 Revised  December 2012 Published  September 2013

This note describes three recent factorizations of banded invertible infinite matrices
  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.
Keywords: factorization., Banded matrix, elimination, banded inverse, doubly infinite matrix.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Gilbert Strang. Three steps on an open road. Inverse Problems & Imaging, 2013, 7 (3) : 961-966. doi: 10.3934/ipi.2013.7.961
References:
[1]

E. Asplund, Inverses of matrices {$a_{ij}$} which satisfy $a_{ij}=0$ for $j > i + p$, Math. Scand., 7 (1959), 57-60.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Lindner, "Infinite Matrices and Their Finite Sections. An Introduction to the Limit Operator Method," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[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.  Google Scholar

[11]

G. Panova, Factorization of banded permutations, Proceedings Amer. Math. Soc., 140 (2012), 3805-3812. doi: 10.1090/S0002-9939-2012-11411-X.  Google Scholar

[12]

J. Plemelj, Riemannsche Funktionenscharen mit gegebener Monodromiegruppe, Monat. Math. Phys., 19 (1908), 211-245. doi: 10.1007/BF01736697.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. Roch, Finite sections of band-dominated operators, AMS Memoirs, 191 (2008). doi: 10.1090/memo/0895.  Google Scholar

[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.  Google Scholar

[18]

G. Strang, Fast transforms: Banded matrices with banded inverses, Proc. Natl. Acad. Sci., 107 (2010), 12413-12416. doi: 10.1073/pnas.1005493107.  Google Scholar

[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.  Google Scholar

[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), 637-646. doi: 10.1137/S0036144503434381.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Lindner, "Infinite Matrices and Their Finite Sections. An Introduction to the Limit Operator Method," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[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.  Google Scholar

[11]

G. Panova, Factorization of banded permutations, Proceedings Amer. Math. Soc., 140 (2012), 3805-3812. doi: 10.1090/S0002-9939-2012-11411-X.  Google Scholar

[12]

J. Plemelj, Riemannsche Funktionenscharen mit gegebener Monodromiegruppe, Monat. Math. Phys., 19 (1908), 211-245. doi: 10.1007/BF01736697.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. Roch, Finite sections of band-dominated operators, AMS Memoirs, 191 (2008). doi: 10.1090/memo/0895.  Google Scholar

[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.  Google Scholar

[18]

G. Strang, Fast transforms: Banded matrices with banded inverses, Proc. Natl. Acad. Sci., 107 (2010), 12413-12416. doi: 10.1073/pnas.1005493107.  Google Scholar

[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.  Google Scholar

[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), 637-646. doi: 10.1137/S0036144503434381.  Google Scholar

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