Cayley-Menger Determinant

Author

Eric W. Weisstein
March 14, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/LinearAlgebra/Cayley-MengerDeterminant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Cayley-MengerDeterminant.html.

©2022 Wolfram Research, Inc. except for portions noted otherwise

Code

V5.2

<<LinearAlgebra`MatrixManipulation`;

CayleyMenger[n_]:=Module[{
u=UpperDiagonalMatrix[d,n+1]^2/.d[i_,i_]:>0,
z=Prepend[Table[1,{n+1}],0]
},
AppendRows[List/@z,AppendColumns[{Rest[z]},u+Transpose[u]]]
]

NormalizedCayleyMenger[n_]:=CayleyMenger[n](-1)^(n+1)/2^n/(n!)^2

V6

CayleyMenger[a_,n_]:=SymmetrizedArray[{{i_,i_}->0,{i_,1}->1,{1,i_}->1,{i_,j_}->((a[i-1,j-1])^2)},{n+2,n+2},Symmetric[All]]

NormalizedCayleyMenger[a_,n_]:=CayleyMenger[a,n](-1)^(n+1)/2^n/(n!)^2

Prefactors

A055546

1/Table[((((-1))^(n+1))/((2^n) (((n!))^2))),{n,0,10}]

FindSequenceFunction[%,n+1]//FullSimplify

n==2

CayleyMenger[d,2]//MatrixForm//TraditionalForm

Normal[CayleyMenger[d,2]]/.Thread[(d@@@Subsets[Range[3],{2}])->Reverse[{a,b,c}]]//TraditionalForm

n==3

CayleyMenger[d,3]//MatrixForm

Normal[CayleyMenger[d,3]]/.d[k__]:>Subscript[d,k]//TraditionalForm

Expand[Det[CayleyMenger[d,3]]/2]

Buchholtz (1992)

m=(({{-d^2, d^2, d^2+b^2, 0, 0, 0}, {e^2-c^2, -e^2, e^2, 0, 0, 0}, {f^2, f^2-a^2, -f^2, 0, 0, 0}, {0, 0, 0, -a^2, a^2, a^2-b^2}, {0, 0, 0, b^2-c^2, -b^2, b^2}, {0, 0, 0, c^2, c^2-a^2, -c^2}}));
v={a,b,c,d,e,f}^2;

Expand[v.m.v]

d1=Det[CayleyMenger[3]]

d2=Expand[2v.m.v/.Thread[{a,b,c,d,e,f}->dd@@@{{2,3},{1,3},{1,2},{1,4},{2,4},{3,4}}]/.dd->d]

d1-d2