It is well known that irreducibly diagonally- dominant matrices with positive diagonal and non-positive off-diagonal elements have positive inverses. A whole class of symmetric circulant and symmetric quindiagonal Toeplitz matrices with positive inverses which do not satisfy the above conditions is found

Meek, D S

English

Brunel University Research Archive

                        TR/50 JANUARY  1975. A  NEW  CLASS  OF  MATRICES WITH  POSITIVE  INVERSES. BY D.  S.  MEEK. This  work was  supported by  a  NATO  postdoctorate  fellowship.        A B S T R A C T  It  is  well  known  that  irreducibly  diagonally- dominant  matrices  with  positive  diagonal  and non-positive  off-diagonal  elements  have  positive inverses.    A  whole  class  of  symmetric  circulant and  symmetric  quindiagonal  Toeplitz  matrices  with positive  inverses  which  do  not  satisfy  the  above conditions  is  found. -   2  - IntroductionA  condition  that  the   zeros  of  a  certain  polynomial  are  positive implies  that  the  corresponding  quindiagonal  matrix  has  a  positive inverse.     A  similar  condition  means  that  the  related  circulant  with five  entries  per  row  has  a  positive  inverse.     The  quindiagonal  matrix is  analysed  by  expressing  the  first  row  of  its   inverse  as  the solution  of  a  finite  difference  equation  and  showing  that  this solution  is  positive.     A  result  of  Trench   (1964)  may  then  "be  used  to prove  that   for  symmetric   quindiagonal  Toeplitz  matrices,   the  complete inverse  is  positive  if  the  first  row  of  the  inverse  is  positive.   The behaviour  of  the  first  row  of  the  inverse  for  very  large  orders  is considered  and  examples  of  this  new  class  for  matrices  are  given. Main  TheoremsTheorem  1.        (Proof  in  section  3) If  the  real  numbers  a  and  b,  b  non-zero,  are  chosen  so  that the  symmetric  polynomial bx4  +  ax3  +  x2  +  ax  +  b (l) has  real  positive  zeros,  then  the  inverse  of  the  symmetric quindiagonal  Toeplitz  matrix,  ,1abababa1abba1aba1Tnxnn⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛= consists   of  positive  elements   for  all  order  n.  - 3 -  The  values  of  a  and  b  for  which  the  polynomial   (l)   has   real positive   zeros   may   be   described  as   follows.   Suppose  the  real   zeror. of  polynomial   (l)   are  r1  ,   l/r1   , r2     and  l/r2    where  r1     and  r2     are positive,  then  the  values  a  and  b  may  be  expressed  as ,2)r/1r()r/1r()r/1r()r/1r(a22112211+++++++−=and .2)r/1r()r/1r(1b2211 ++++= It  is  easy  to  verify  that    -  2/3  <  a  <  0  and  0  <  b ≤  1/6. The  relation  a  +  b  ≥  ½   follows  from  the  inequality  (r1  +l/r1   -2)(r2  +l/r2   - 2 )   ≥  0  and  a2  + 8b2 - 4 b  ≥  0   is  the  condition that  r1  and  r2  are  real. The  above  inequalities  describe  a  region,  R, in  the  a  - b   plane  bounded  by  the  three  curves  b  =  0,   a +b  = -21   and a2 + 8b2 -4b = 0 with intersection  points   (0,0),(-1/2,0)  and (-  2/3,  1/6).  Theorem  2. If  the  real  numbers   a  and  b,   b  non-zero,   are   chosen  such  that the  symmetric  polynomial bx4 + ax3 + x2 + ax + b    (l) has  real  positive  zeros,  not  equal  to  one,  then  the  inverse of  the  symmetric  circulant  matrix Cn = circ  (l,a,b,0   ...  , 0,b,a)nxn   (3) consists  of  positive  entries  for  all  orders  n.    — 4 –  Proof  The matrix Cn  ,   in  (3),   can be  factored into the product  of two matrices, ,])PP(I)r/r([])PP(I)r/r([C n122111 11−− +−++−+= where    P    is  the  permutation  matrix  circ   (0,1,0,...,0)nxn  and r1 and r2 are  zeros  of  (l)   not  equal  to  1.  However,  both  factors are  irreducibly  diagonally  dominant  and  have  positive  inverses, Varga  (1962),  and  consequently,  the  inverse  of  the  matrix  Cn    is positive. The  above  method  of  proof  may  be  used  for  a more  general theorem concerning  circulant  matrices,  Meek  (1973).  Theorem  3.   (Trench(1964)). If  Tn  is the  nth order,  n  ≥  3,  symmetric  Toeplitz  matrix  in  (2) and  the  first  row  of  the  matrix  is  positive,  then  11+−nT.nTwith 01 >−the  matrix .T  n 011 >+−Proof  The matrix  may  be  partitioned  in  the  form 11+−nT11111+×+⎥⎥⎦⎤⎢⎢⎣⎡+−=+−nnTnnnnt/WWnTWTWtnT  If  the  first  row  of   is  positive,  then  both  t  and  the  lxn .Tn 011 >+−Vector WnT are  positive.    However, thus   Tn-1> 0 ,  all  of  the  elements of  the  matrix  are  positive. 11+−nT-  5  -   3.     Proof  of  Theorem  1. The  first  row  of  the  matrix nT1− is  shown  positive  by  solving  a difference  equation  so  that  theorem  3  is  an  induction  step  in  the proof  of  theorem  1.     It  is   easy  to  verify  that   031 >−T  when  a  and  b are  chosen  so  that   (a,b)   is  in  region  R. The  difference  equation  for  the  elements  of  the  first  row  of  the    isnTmatrix1−bD r-2  +   aDr-1  ,  + Dr + aDr+1 +  bDr+2 =  er ,r = 1,2,... ,n,      (4) with  end  conditions  D-1 = D0 = Dn+1 = Dn+2 = 0, where ⎩⎨⎧===.n,......,3,2r01r1erA  particular  solution  to  equation  (4)   is  the  function ⎩⎨⎧= +=−=,2n,...,1,0r01rb/1rH  while  the  general  solution  falls   into  four  cases,  depending  upon the  zeros  of  the  polynomial   (l).  The  more  general  case  of  positive real  distinct  zeros  not  equal  to  1  will  be  discussed  first. The  zeros  of the  polynomial   (l)   are  positive,  distinct  and not  equal  to  1,   if  and  only  if  a  and  b  are  such  that  the  point   (a,b) lies  in  the  interior  of  the  region  R.    For  convenience  take r1   > r2 >  1   to  be  zeros  of  (l),  then  the  general  solution  to  the difference  equation ( 4 )   is  of  the  form .n......,,,r,HrDrrCrrBrrArD rr 2012211 +−=−++−+−= -  6  - If  the  function  f(r,n)   is  defined )5(,tstststststs)n,r(f 2n1rn2rn1n1n2rn1rn2nr11r ++−+−+++−+−+= + + − − +  ,rrtandrrswhere 2211ω−−ω≡ω−−ω≡ ωω    then the solution satisfy- ing the end condition is                                    Dr    =  f(r,n)/  (b  f(l,n+l))   +  Hr    , r =-1,0,... ,n+2.              (6) As    n    becomes   large,  Dr    approaches  the  function ⎩⎨⎧≤<==∞→ ,tt)r,br/(Dnlimr 10001 21where    r ≡(n-l)t+l. The substitution r1 = e(θ+ ψ )  ,  r2  =  e(θ+ ψ )   ,    where θ >  ψ  >  0,  transforms  the  function  f(r,n)   in  (5)    into f(r,n)  =l6sinh(n+l)θ  sinh(n-r+2)θ   sinhrψ   sinh  ψ-  l6sinh(n+l)ψ   sinh(n-r+2)ψ   sinhrθ  sinhθ which  is  positive  for  r = 1,2,...,n   since  both sinh(n+l)θ  sinhrψ   -  sinh(n+l)ψ   sinhrθ and sinh(n-r+2)θ  sinhψ  -  sinh(n-r+2)ψ  sinhe are positive r = 1,2,... ,n, (see the appendix) - Now tooth  b  and f(r,n) are  positive, thus Dr , r= 1,2,...,n  in equation (6) is positive.       The  results  for  the  remaining  three  cases  may  now  be summarized. The  polynomial  (l)  may  have  one  repeated  zero,  that  is  the  zeros  are r1, 1/r1,l, l,r1 >1, and a+b  = -1/2, or it  may have two pairs  of repeated zeros, that is the zeros are r1 , 1/r1 , r1 ,1/r1 ,r1 >1 and - 7 –    a2 + 8b2 - 4b = 0 , of  it. may  have all.   Four  zeros   equal to 1, whence a= -2/3   b = 1/ 6 .  The solulion of the difference equation   (4)   is  of the same form as   in  equation   (6)  with the  function f(r,n)  being defined f(r,n)   = sr+rs1+ ( n-r+1)sn+2-( n+1) s n-r+2-(n-r+2)sn+1+(n+2)sn-r+1, f(r,n)  = rsn+1sn-r+2-(n+1)(n-r+2) srs1   , and f(r,n) = r(n-r+l)(n-r+2) , respectively.    As  n  becomes  large,  Dr  ,  in  these  three  cases  approaches the  functions  ⎩⎨⎧≤<−−==∞→ ,t))r(b/()t(t)br/(Dnlimr 10110111  ⎩⎨⎧≤<==∞→ ,tt)br/(Dnlimr 10001 12  and ⎩⎨⎧≤<−==∞→ ,t)t(nttDnlimr 1016062   where  r = ( n - l ) t + l .  The  substitution  r1   =  e2θ  ,θ  > 0,  in  the  first  case  yields f(r,n)  =8sinh(n-r+2)θ  sinhθ[(n+l)cosh(n+l)θ    sinhrθ-rcoshrθ     sinh(n+l)θ] +  8sinh(n+l)θ     sinhr9 [ (n-r+2)cosh(n-r+2)θ  sinhθ-coshθ  sinh(n-r+2)θ] in which  both  of  the  terms  are  positive  for  each  n  and  r=l,2,...,n (see  the  appendix).  The substitution  r1 = eθ  ,θ  >  0,  in  the  second  case   -  8  -    gives f(r,n)  = 4[rsinh(n+l)θ]:sinh(n-r+2)θ]-U[(n+l)sinhre3L(n-p+2)sinhe] which  is  also  positive  for  each  n  and  r=l,2,—  n    (see  the  appendix). In  the  third  case,   f(r,n)   is  ohviously  positive  for  each  n  and r  =  1,2,...,  n  . 4.     ExamplesThe  matrix  arising  from  quintic  polynomial  spline  interpolation on  a  uniform partition  with  non-periodic  boundary  conditions  is  the quindiagonal ,662612611662612666A⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎝⎛=Ahlberg,  Nilson  and  Walsh,   p. 124   (1967).   The  related  matrix  DADT  , where  D  is  the  diagonal  matrix  diag  ( 1,-1,...,(-1) n-1) nxn  ,  has  a positive  inverse  since  the  zeros   of 1/66 z4 -  26/66 z3  +  z2 - 26 /66 z + 1/66 are  real  and  positive Hoskins  and  Ponzo   (1972)  have  found  another  class  of  symmetric Toeplitz  matrices  with  positive  inverses  which  intersects  with  the class   described  here  in  the  nxn  matrix  ⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎝⎛−−−−13/26/13/26/16/113/26/13/21 -   9  -   6.      Appendix Lemma    1. The  function   (sinhmθ)m  is  greater  than  θ   and  monotone  increasing in  m  when  θ > 0, m > 0. Proof The  result  follows  from  a  Maclaurin  expansion  of  sinhmθ . Lemma  2. The  function  m  cothmθ   is  greater  than  θ   and  is  monotone increasing  in  m, θ > 0, m > 0. Proof The  derivative  of  m  cothm9  with  respect  to  m  can  be  shown  to be  positive  using  lemma  1.   for  θ  >  0,  m  >  0  and     (m  cothmθ)  =  0mlim→θ.    Lemma  3. The  function  (sinhmθ )/(sinhmψ )   is  greater  than  θ/   and  ψincreases with  m,   θ > 0, ψ   >  0,  m > 0. The  derivative  of   (sinhmθ)/(sinhimψ )  with  respect  to  m  can  be shown  to  be  positive  using  lemma  2.   and      (sinhmθ)/(sinhimy)   =  0mlim→θ/ψ .  -   10  - References1. Ahlbcrg,  J.H.,  Wilson,   E.N.   and  Walsh,  J.L.     "The  Theory  of  Splines and  Their  Applications".     Academic  Press,New  York,1967. 2. Hoskins,  W.D.,  and  Ponzo,  P.J.     Some  properties  of  a  class  of  band matrices.  Math.Comp.118(1972),pp.393-400. 3. Meek,  D.S.   "On  the  Numerical  Construction  and  Approximation  of Some  Piecewise  Polynomial  Functions".  Ph.D.thesis,  University of  Manitoba,Canada.1973. 4.  Trench,  W.J.    An  algorithm  for  the  inversion  of  finite  Toeplitz matrices.  Journal  of  S.I.A.M.3(1964). 5.      Varga,  R.S.   "Matrix  Iterative  Analysis".  Prentice-Hall,  Englewood  Cliffs, New  Jersey,  1962.             XB 2356843 7  

An algorithm for the inversion of finite Toeplitz matrices.

A new class of matrices with positive inverses

Abstract

Similar works

Full text

Available Versions

Brunel University Research Archive