OFFSET
1,4
COMMENTS
The 4 X 4 is a g(u,v) type of matrix where Minkowski is n(u,v)={1,1,1,-1}: single hyperbolic index. Matrices: 1 X 1 {{-1}} 2 X 2 {{1, -1}, {-1, -1}} 3 X 3 {{1, -1, 0}, {-1, 1, -1}, {0, -1, -1}} 4 X 4 {{1, -1, 0, 0}, {-1, 1, -1, 0}, {0, -1, 1, -1}, {0, 0, -1, -1}} 5 X 5 {{1, -1, 0, 0, 0}, {-1, 1, -1, 0, 0}, {0, -1, 1, -1, 0}, {0, 0, -1, 1, -1}, {0, 0, 0, -1, -1}
FORMULA
m(n,m,d)=If[ n == m && n < d && m < d, 1, If[n == m - 1 || n == m + 1, -1, If[n == m == d, -1, 0]]
EXAMPLE
Triangular sequence:
{{-1}},
{-1, -1},
{-2, 0, 1},
{-1, 3, 1, -1},
{1, 4, -3, -2,1},
{2, 0, -8, 2, 3, -1},
{1, -6, -5, 12, 0, -4, 1},
{-1, -7, 9, 15, -15, -3, 5, -1},
{-2, 0, 21, -6, -30, 16, 7, -6, 1},
MATHEMATICA
T[n_, m_, d_] := If[ n == m && n < d && m < d, 1, If[n == m - 1 || n == m + 1, -1, If[n == m == d, -1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], { d, 1, 10}]] Flatten[a] MatrixForm[a]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Nov 02 2006
STATUS
approved