1,5

Row sums are:

{1,1, 0, -1, 2, 0, -2, 3, 0, -3, 4, 0}

Testing for orthogonalization:

(* make equal length vectors*)

f[n_] := Table[0, {i, 1, n}];

b = Table[Join[a[[n]], f[Length[a] - n]], {n, 1, Length[a]}];

(* apply the Gram - Schmidt orthogonalization function*)

<< LinearAlgebra`Orthogonalization`;

c = N[GramSchmidt[b]]

This procedure gives :

{{1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.},

{0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.},

{0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, < -"Null vector"

{0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.}, < -"Null vector"

{0., 0., 0.447214, 0.894427, 0., 0., 0., 0., 0., 0., 0.},

{0., 0., -0.26968, 0.13484, 0.6742,0.6742, 0., 0., 0., 0., 0.},

{0., 0., 0.0772091, -0.0386046, -0.617673, 0.656278, 0.42465, 0., 0., 0., 0.},

{0., 0., 0.387147, -0.193574, -0.14518, 0.338754, -0.822688, 0., 0., 0., 0.},

{0., 0., 0.755929, -0.377964, 0.377964, 0., 0.377964, 0., 0., 0., 0.},

{0., 0., 0., 0., 0., 0., 0., 0.707107, 0.707107, 0., 0.},

{0., 0., 0., 0., 0.,0., 0., -0.597614, 0.597614, 0.478091, 0.239046}}

This seem to indicate that pentadiagonal matrices can represent a

higher form of orthogonality

than is considered in traditional Hilbert space theory.

Biorthogonal matrices are the first concrete example that is actually in use

in engineering applications that I know of.

Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.

h(i,j)=If(i=j,1, If( i-j+/-1=0,-1, If( i-j+/-2=0,1, Else 0),i.j<=n

{1},

{-1, 1},

{1, -2, 0},

{-1, 3, 0,0},

{1, -4, 1, 2, 0},

{-1, 5, -3, -5, 1, 1},

{1, -6, 6, 8, -5, -2, 1},

{-1, 7, -10, -10, 14, 4, -4, 0},

{1, -8, 15, 10, -29, -4, 12, 0,0},

{-1, 9, -21, -7, 50, -4, -30, 4, 4, 0},

{1, -10, 28, 0, -76, 28, 61, -20, -15, 2, 1}

Clear[M, T, MO, a, b, p, Ox, Oxt, n, m, d, c] a[n_] := 1; b[n_] := -1; c[n_] := 1; (* symmetrical Pentadiagonal matrices on sequences a[n], b[n], c[n]*) (* where p[x, 0] = 1; p[x, -1] = 0; p[x, j + 1] = (x - a[j + 1])*p[x, j] - b[j]*p[x, j - 1]*) (* reference : A First Course in Numerical Analysis, page 256, 1978, ISBN 0070511586, Anthony Ralston and Philip Rabinowitz*) (* note : b[n] is squared*) T[n_, m_, d_] := If[ n == m, a[n], If[n == m - 1 || n == m + 1 || n == m - 2 || n == m + 2, If[n == m - 1, b[m - 1], If[ n == m + 1, b[n - 1], If[n == m - 2 || n == m + 2, If[n == m - 2, c[m - 2], If[n == m + 2, c[n - 2], 0]]]]]]] MO[d_] := Table[If[TrueQ[T[n, m, d] == Null], 0, T[n, m, d]], {n, 1, d}, {m, 1, d}]; a = Join[{{1}}, Table[Reverse[CoefficientList[CharacteristicPolynomial[MO[n], x], x]], {n, 1, 10}]];

Adjacent sequences: A136490 A136491 A136492 this_sequence A136494 A136495 A136496

Sequence in context: A127701 A004199 A062283 this_sequence A132213 A119900 A141097

uned,tabl,sign

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 21 2008