CR - 9D

The 9-D Case

We obtain the 9-D convergent ratio vector by successive transformations of a base vector by the 9-D triangular matrix:

0	0	0	0	0	0	0	0	1		1
0	0	0	0	0	0	0	1	1		2
0	0	0	0	0	0	1	1	1		3
0	0	0	0	0	1	1	1	1		4
0	0	0	0	1	1	1	1	1	×	5
0	0	0	1	1	1	1	1	1		6
0	0	1	1	1	1	1	1	1		7
0	1	1	1	1	1	1	1	1		8
1	1	1	1	1	1	1	1	1		9

This leads to the sequence

9-D Sequence
1	9	45	285	1695	10317	62349	377739	2286648		s₁
2	17	89	561	3345	20349	123003	745161	4510947		s₂
3	24	131	820	4906	29820	180312	1092234	6612243		s₃
4	30	170	1055	6336	38471	232715	1409487	8533227		s₄
5	35	205	1260	7596	46067	278782	1688269	10221496	…	s₅
6	39	235	1430	8651	52403	317253	1920984	11630983		s₆
7	42	259	1561	9471	57309	347073	2101296	12723217		s₇
8	44	276	1650	10032	60654	367422	2224299	13468378		s₈
9	45	285	1695	10317	62349	377739	2286648	13846117		s₉

The Matrix Aspect

The 9-D convergent ratio includes 36 relations between components. Together with inverses this creates 72 core matrices

Table of 9-D Core Matrices

s₁→s₉ (s₉/s₁)
0	0	0	0	0	0	0	0	1
0	0	0	0	0	0	0	1	1
0	0	0	0	0	0	1	1	1
0	0	0	0	0	1	1	1	1
0	0	0	0	1	1	1	1	1
0	0	0	1	1	1	1	1	1
0	0	1	1	1	1	1	1	1
0	1	1	1	1	1	1	1	1
1	1	1	1	1	1	1	1	1

s₁→s₈ (s₈/s₁)
0	0	0	0	0	0	0	1	0
0	0	0	0	0	0	1	0	1
0	0	0	0	0	1	0	1	1
0	0	0	0	1	0	1	1	1
0	0	0	1	0	1	1	1	1
0	0	1	0	1	1	1	1	1
0	1	0	1	1	1	1	1	1
1	0	1	1	1	1	1	1	1
0	1	1	1	1	1	1	1	1

s₁→s₇ (s₇/s₁)
0	0	0	0	0	0	1	0	0
0	0	0	0	0	1	0	1	0
0	0	0	0	1	0	1	0	1
0	0	0	1	0	1	0	1	1
0	0	1	0	1	0	1	1	1
0	1	0	1	0	1	1	1	1
1	0	1	0	1	1	1	1	1
0	1	0	1	1	1	1	1	1
0	0	1	1	1	1	1	1	1

s₁→s₆ (s₆/s₁)
0	0	0	0	0	1	0	0	0
0	0	0	0	1	0	1	0	0
0	0	0	1	0	1	0	1	0
0	0	1	0	1	0	1	0	1
0	1	0	1	0	1	0	1	1
1	0	1	0	1	0	1	1	1
0	1	0	1	0	1	1	1	1
0	0	1	0	1	1	1	1	1
0	0	0	1	1	1	1	1	1

s₁→s₅ (s₅/s₁)
0	0	0	0	1	0	0	0	0
0	0	0	1	0	1	0	0	0
0	0	1	0	1	0	1	0	0
0	1	0	1	0	1	0	1	0
1	0	1	0	1	0	1	0	1
0	1	0	1	0	1	0	1	1
0	0	1	0	1	0	1	1	1
0	0	0	1	0	1	1	1	1
0	0	0	0	1	1	1	1	1

s₁→s₄ (s₄/s₁)
0	0	0	1	0	0	0	0	0
0	0	1	0	1	0	0	0	0
0	1	0	1	0	1	0	0	0
1	0	1	0	1	0	1	0	0
0	1	0	1	0	1	0	1	0
0	0	1	0	1	0	1	0	1
0	0	0	1	0	1	0	1	1
0	0	0	0	1	0	1	1	1
0	0	0	0	0	1	1	1	1

s₁→s₃ (s₃/s₁)
0	0	1	0	0	0	0	0	0
0	1	0	1	0	0	0	0	0
1	0	1	0	1	0	0	0	0
0	1	0	1	0	1	0	0	0
0	0	1	0	1	0	1	0	0
0	0	0	1	0	1	0	1	0
0	0	0	0	1	0	1	0	1
0	0	0	0	0	1	0	1	1
0	0	0	0	0	0	1	1	1

s₁→s₂ (s₂/s₁)
0	1	0	0	0	0	0	0	0
1	0	1	0	0	0	0	0	0
0	1	0	1	0	0	0	0	0
0	0	1	0	1	0	0	0	0
0	0	0	1	0	1	0	0	0
0	0	0	0	1	0	1	0	0
0	0	0	0	0	1	0	1	0
0	0	0	0	0	0	1	0	1
0	0	0	0	0	0	0	1	1

s₉→s₁ (s₁/s₉)
0	0	0	0	0	0	0	-1	1
0	0	0	0	0	0	-1	1	0
0	0	0	0	0	-1	1	0	0
0	0	0	0	-1	1	0	0	0
0	0	0	-1	1	0	0	0	0
0	0	-1	1	0	0	0	0	0
0	-1	1	0	0	0	0	0	0
-1	1	0	0	0	0	0	0	0
1	0	0	0	0	0	0	0	0

s₉→s₂ (s₂/s₉)
0	0	0	0	0	0	-1	1	0
0	0	0	0	0	-1	1	-1	1
0	0	0	0	-1	1	-1	1	0
0	0	0	-1	1	-1	1	0	0
0	0	-1	1	-1	1	0	0	0
0	-1	1	-1	1	0	0	0	0
-1	1	-1	1	0	0	0	0	0
1	-1	1	0	0	0	0	0	0
0	1	0	0	0	0	0	0	0

s₉→s₃ (s₃/s₉)
0	0	0	0	0	-1	1	0	0
0	0	0	0	-1	1	-1	1	0
0	0	0	-1	1	-1	1	-1	1
0	0	-1	1	-1	1	-1	1	0
0	-1	1	-1	1	-1	1	0	0
-1	1	-1	1	-1	1	0	0	0
1	-1	1	-1	1	0	0	0	0
0	1	-1	1	0	0	0	0	0
0	0	1	0	0	0	0	0	0

s₉→s₄ (s₄/s₉)
0	0	0	0	-1	1	0	0	0
0	0	0	-1	1	-1	1	0	0
0	0	-1	1	-1	1	-1	1	0
0	-1	1	-1	1	-1	1	-1	1
-1	1	-1	1	-1	1	-1	1	0
1	-1	1	-1	1	-1	1	0	0
0	1	-1	1	-1	1	0	0	0
0	0	1	-1	1	0	0	0	0
0	0	0	1	0	0	0	0	0

s₉→s₅ (s₅/s₉)
0	0	0	-1	1	0	0	0	0
0	0	-1	1	-1	1	0	0	0
0	-1	1	-1	1	-1	1	0	0
-1	1	-1	1	-1	1	-1	1	0
1	-1	1	-1	1	-1	1	-1	1
0	1	-1	1	-1	1	-1	1	0
0	0	1	-1	1	-1	1	0	0
0	0	0	1	-1	1	0	0	0
0	0	0	0	1	0	0	0	0

s₉→s₆ (s₆/s₉)
0	0	-1	1	0	0	0	0	0
0	-1	1	-1	1	0	0	0	0
-1	1	-1	1	-1	1	0	0	0
1	-1	1	-1	1	-1	1	0	0
0	1	-1	1	-1	1	-1	1	0
0	0	1	-1	1	-1	1	-1	1
0	0	0	1	-1	1	-1	1	0
0	0	0	0	1	-1	1	0	0
0	0	0	0	0	1	0	0	0

s₉→s₇ (s₇/s₉)
0	-1	1	0	0	0	0	0	0
-1	1	-1	1	0	0	0	0	0
1	-1	1	-1	1	0	0	0	0
0	1	-1	1	-1	1	0	0	0
0	0	1	-1	1	-1	1	0	0
0	0	0	1	-1	1	-1	1	0
0	0	0	0	1	-1	1	-1	1
0	0	0	0	0	1	-1	1	0
0	0	0	0	0	0	1	0	0

s₉→s₈ (s₈/s₉)
-1	1	0	0	0	0	0	0	0
1	-1	1	0	0	0	0	0	0
0	1	-1	1	0	0	0	0	0
0	0	1	-1	1	0	0	0	0
0	0	0	1	-1	1	0	0	0
0	0	0	0	1	-1	1	0	0
0	0	0	0	0	1	-1	1	0
0	0	0	0	0	0	1	-1	1
0	0	0	0	0	0	0	1	0

s₈→s₃ (s₃/s₈)
0	0	0	0	0	0	-1	0	1
0	0	0	0	0	-1	0	0	1
0	0	0	0	-1	0	0	1	0
0	0	0	-1	0	0	1	0	0
0	0	-1	0	0	1	0	0	0
0	-1	0	0	1	0	0	0	0
-1	0	0	1	0	0	0	0	0
0	0	1	0	0	0	0	0	0
1	1	0	0	0	0	0	0	0

s₇→s₅ (s₅/s₇)
0	0	0	0	0	-1	0	1	0
0	0	0	0	-1	0	0	0	1
0	0	0	-1	0	0	0	0	1
0	0	-1	0	0	0	0	1	0
0	-1	0	0	0	0	1	0	0
-1	0	0	0	0	1	0	0	0
0	0	0	0	1	0	0	0	0
1	0	0	1	0	0	0	0	0
0	1	1	0	0	0	0	0	0

s₆→s₇ (s₇/s₆)
0	0	0	0	-1	0	1	0	0
0	0	0	-1	0	0	0	1	0
0	0	-1	0	0	0	0	0	1
0	-1	0	0	0	0	0	0	1
-1	0	0	0	0	0	0	1	0
0	0	0	0	0	0	1	0	0
1	0	0	0	0	1	0	0	0
0	1	0	0	1	0	0	0	0
0	0	1	1	0	0	0	0	0

s₅→s₉ (s₉/s₅)
0	0	0	-1	0	1	0	0	0
0	0	-1	0	0	0	1	0	0
0	-1	0	0	0	0	0	1	0
-1	0	0	0	0	0	0	0	1
0	0	0	0	0	0	0	0	1
1	0	0	0	0	0	0	1	0
0	1	0	0	0	0	1	0	0
0	0	1	0	0	1	0	0	0
0	0	0	1	1	0	0	0	0

s₄→s₈ (s₈/s₄)
0	0	-1	0	1	0	0	0	0
0	-1	0	0	0	1	0	0	0
-1	0	0	0	0	0	1	0	0
0	0	0	0	0	0	0	1	0
1	0	0	0	0	0	0	0	1
0	1	0	0	0	0	0	0	1
0	0	1	0	0	0	0	1	0
0	0	0	1	0	0	1	0	0
0	0	0	0	1	1	0	0	0

s₃→s₆ (s₆/s₃)
0	-1	0	1	0	0	0	0	0
-1	0	0	0	1	0	0	0	0
0	0	0	0	0	1	0	0	0
1	0	0	0	0	0	1	0	0
0	1	0	0	0	0	0	1	0
0	0	1	0	0	0	0	0	1
0	0	0	1	0	0	0	0	1
0	0	0	0	1	0	0	1	0
0	0	0	0	0	1	1	0	0

s₂→s₄ (s₄/s₂)
-1	0	1	0	0	0	0	0	0
0	0	0	1	0	0	0	0	0
1	0	0	0	1	0	0	0	0
0	1	0	0	0	1	0	0	0
0	0	1	0	0	0	1	0	0
0	0	0	1	0	0	0	1	0
0	0	0	0	1	0	0	0	1
0	0	0	0	0	1	0	0	1
0	0	0	0	0	0	1	1	0

s₅→s₄ (s₄/s₅)
1	0	0	0	0	0	0	1	-1
0	1	0	0	0	0	1	-1	0
0	0	1	0	0	1	-1	0	0
0	0	0	1	1	-1	0	0	0
0	0	0	1	0	0	0	0	0
0	0	1	-1	0	1	0	0	0
0	1	-1	0	0	0	1	0	0
1	-1	0	0	0	0	0	1	0
-1	0	0	0	0	0	0	0	1

s₄→s₇ (s₇/s₄)
1	0	0	0	0	0	-1	0	1
0	1	0	0	0	-1	0	0	1
0	0	1	0	-1	0	0	1	0
0	0	0	0	0	0	1	0	0
0	0	-1	0	1	1	0	0	0
0	-1	0	0	1	1	0	0	0
-1	0	0	1	0	0	1	0	0
0	0	1	0	0	0	0	1	0
1	1	0	0	0	0	0	1	1

s₃→s₉ (s₉/s₃)
1	0	0	0	-1	0	1	0	0
0	1	0	-1	0	0	0	1	0
0	0	0	0	0	0	0	0	1
0	-1	0	1	0	0	0	0	1
-1	0	0	0	1	0	0	1	0
0	0	0	0	0	1	1	0	0
1	0	0	0	0	1	1	0	0
0	1	0	0	1	0	0	1	0
0	0	1	1	0	0	0	0	1

s₂→s₆ (s₆/s₂)
1	0	-1	0	1	0	0	0	0
0	0	0	0	0	1	0	0	0
-1	0	1	0	0	0	1	0	0
0	0	0	1	0	0	0	1	0
1	0	0	0	1	0	0	0	1
0	1	0	0	0	1	0	0	1
0	0	1	0	0	0	1	1	0
0	0	0	1	0	0	1	1	0
0	0	0	0	1	1	0	0	1

s₇→s₂ (s₂/s₇)
-1	0	0	-1	0	1	0	0	0
0	-1	-1	0	0	0	1	0	0
0	-1	-1	0	0	0	0	1	0
-1	0	0	-1	0	0	0	0	1
0	0	0	0	-1	0	0	0	1
1	0	0	0	0	-1	0	1	0
0	1	0	0	0	0	0	0	0
0	0	1	0	0	1	0	-1	0
0	0	0	1	1	0	0	0	-1

s₈→s₅ (s₅/s₈)
-1	-1	0	1	0	0	0	0	0
-1	-1	0	0	1	0	0	0	0
0	0	-1	0	0	1	0	0	0
1	0	0	-1	0	0	1	0	0
0	1	0	0	-1	0	0	1	0
0	0	1	0	0	-1	0	0	1
0	0	0	1	0	0	-1	0	1
0	0	0	0	1	0	0	0	0
0	0	0	0	0	1	1	0	-1

s₃→s₇ (s₇/s₃)
0	-1	0	1	0	0	0	-1	1
-1	0	0	0	1	0	-1	1	0
0	0	0	0	0	0	1	0	0
1	0	0	0	-1	1	1	0	0
0	1	0	-1	1	0	0	1	0
0	0	0	1	0	0	0	0	1
0	-1	1	1	0	0	0	0	1
-1	1	0	0	1	0	0	1	0
1	0	0	0	0	1	1	0	0

s₃→s₅ (s₅/s₃)
-1	1	0	-1	1	0	0	1	-1
1	-1	0	1	-1	1	1	-1	0
0	0	0	0	1	0	0	0	0
-1	1	0	0	1	0	-1	1	0
1	-1	1	1	-1	0	1	-1	1
0	1	0	0	0	0	0	1	0
0	1	0	-1	1	0	0	1	0
1	-1	0	1	-1	1	1	-1	1
-1	0	0	0	1	0	0	1	0

s₃→s₄ (s₄/s₃)
1	0	0	0	-1	1	1	-1	0
0	1	0	-1	1	0	0	1	-1
0	0	0	1	0	0	0	0	0
0	-1	1	1	0	0	0	-1	1
-1	1	0	0	1	0	-1	1	0
1	0	0	0	0	0	1	0	0
1	0	0	0	-1	1	1	0	0
-1	1	0	-1	1	0	0	1	0
0	-1	0	1	0	0	0	0	1

s₄→s₆ (s₆/s₄)
0	1	1	0	-1	-1	0	0	1
1	1	1	0	-1	-1	-1	1	1
1	1	0	0	0	-1	0	0	1
0	0	0	0	0	1	0	0	0
-1	-1	0	0	1	1	1	0	-1
-1	-1	-1	1	1	1	1	0	-1
0	-1	0	0	1	1	0	0	0
0	1	0	0	0	0	0	0	1
1	1	1	0	-1	-1	0	1	1

s₆→s₈ (s₈/s₆)
-1	0	0	-1	0	0	0	1	0
0	-1	-1	0	-1	0	1	0	1
0	-1	-1	-1	0	0	0	1	1
-1	0	-1	-1	0	0	0	1	1
0	-1	0	0	-1	0	1	0	1
0	0	0	0	0	0	0	1	0
0	1	0	0	1	0	0	0	0
1	0	1	1	0	1	0	-1	0
0	1	1	1	1	0	0	0	-1

s₅→s₆ (s₆/s₅)
-1	1	0	0	0	0	0	-1	1
1	-1	1	0	0	0	-1	1	0
0	1	-1	1	0	-1	1	0	0
0	0	1	-1	0	1	0	0	0
0	0	0	0	0	1	0	0	0
0	0	-1	1	1	-1	1	0	0
0	-1	1	0	0	1	-1	1	0
-1	1	0	0	0	0	1	-1	1
1	0	0	0	0	0	0	1	0

s₇→s₈ (s₈/s₇)
1	0	-1	1	1	-1	0	1	-1
0	0	1	0	0	1	0	-1	0
-1	1	1	0	0	1	0	-1	0
1	0	0	1	1	-1	0	1	-1
1	0	0	1	0	0	0	0	0
-1	1	1	-1	0	1	0	-1	1
0	0	0	0	0	0	0	1	0
1	-1	-1	1	0	-1	1	1	0
-1	0	0	-1	0	1	0	0	1

s₈→s₇ (s₇/s₈)
1	1	-1	-1	0	1	1	0	-1
1	0	0	-1	0	1	1	0	-1
-1	0	0	1	0	0	0	0	0
-1	-1	1	1	1	-1	-1	0	1
0	0	0	1	0	0	-1	0	1
1	1	0	-1	0	0	1	0	0
1	1	0	-1	-1	1	1	1	-1
0	0	0	0	0	0	1	0	0
-1	-1	0	1	1	0	-1	0	1

s₆→s₅ (s₅/s₆)
0	1	0	0	1	0	-1	0	0
1	0	1	1	0	0	0	-1	0
0	1	1	1	0	0	0	0	-1
0	1	1	0	1	0	0	0	-1
1	0	0	1	0	1	0	-1	0
0	0	0	0	1	0	0	0	0
-1	0	0	0	0	0	0	-1	0
0	-1	0	0	-1	0	1	0	1
0	0	-1	-1	0	0	0	1	1

s₈→s₆ (s₆/s₈)
0	0	0	-1	0	1	1	0	-1
0	0	-1	0	0	1	1	0	-1
0	-1	0	0	1	0	0	0	0
-1	0	0	1	0	0	-1	0	1
0	0	1	0	0	-1	0	0	1
1	1	0	0	-1	0	0	1	0
1	1	0	-1	0	0	1	0	0
0	0	0	0	0	1	0	0	0
-1	-1	0	1	1	0	0	0	0

s₆→s₄ (s₄/s₆)
0	0	1	1	0	0	0	-1	0
0	1	1	1	1	0	0	-1	-1
1	1	1	1	1	0	0	-1	-1
1	1	1	1	0	1	0	-1	-1
0	1	1	0	1	0	0	0	-1
0	0	0	1	0	0	0	0	0
0	-1	0	0	0	0	0	0	1
-1	0	-1	-1	0	0	0	1	1
0	-1	-1	-1	-1	0	1	1	1

s₄→s₃ (s₃/s₄)
0	0	-1	0	1	1	0	-1	0
0	-1	0	0	1	1	0	0	-1
-1	0	0	1	0	0	1	0	-1
0	0	1	0	0	0	0	0	0
1	1	0	0	0	0	-1	0	1
1	1	0	0	0	-1	0	0	1
0	0	1	0	-1	0	0	1	0
-1	0	0	0	0	0	1	0	0
0	-1	-1	0	1	1	0	0	0

s₅→s₃ (s₃/s₅)
-1	1	0	0	0	0	1	-1	0
1	-1	1	0	0	1	-1	1	-1
0	1	-1	1	1	-1	1	-1	0
0	0	1	0	0	1	-1	0	0
0	0	1	0	0	0	0	0	0
0	1	-1	1	0	-1	1	0	0
1	-1	1	-1	0	1	-1	1	0
-1	1	-1	0	0	0	1	-1	1
0	-1	0	0	0	0	0	1	0

s₂→s₈ (s₈/s₂)
-1	0	1	0	-1	0	1	0	0
0	0	0	0	0	0	0	1	0
1	0	-1	0	1	0	0	0	1
0	0	0	0	0	1	0	0	1
-1	0	1	0	0	0	1	1	0
0	0	0	1	0	0	1	1	0
1	0	0	0	1	1	0	0	1
0	1	0	0	1	1	0	0	1
0	0	1	1	0	0	1	1	0

s₂→s₅ (s₅/s₂)
1	0	-1	0	1	1	-1	-1	1
0	0	0	0	1	0	0	0	0
-1	0	1	1	-1	0	1	1	-1
0	0	1	0	0	0	1	0	0
1	1	-1	0	1	1	-1	0	1
1	0	0	0	1	0	0	0	1
1	0	1	1	-1	0	1	1	0
-1	0	1	0	0	0	1	1	0
1	0	-1	0	1	1	0	0	1

s₂→s₃ (s₃/s₂)
-1	0	1	1	-1	-1	1	1	-1
0	0	1	0	0	0	0	0	0
1	1	-1	0	1	1	-1	-1	1
1	0	0	0	1	0	0	0	0
-1	0	1	1	-1	0	1	1	-1
-1	0	1	0	0	0	1	0	0
1	0	-1	0	1	1	-1	0	1
1	0	-1	0	1	0	0	0	1
-1	0	1	0	-1	0	1	1	0

s₃→s₂ (s₂/s₃)
-1	1	0	0	1	-1	-1	1	0
1	-1	1	1	-1	0	0	-1	1
0	1	0	0	0	0	0	0	0
0	1	0	-1	1	0	0	1	-1
1	-1	0	1	-1	1	1	-1	0
-1	0	0	0	1	0	0	0	0
-1	0	0	0	1	0	-1	1	0
1	-1	0	1	-1	0	1	-1	1
0	1	0	-1	0	0	0	1	0

s₅→s₂ (s₂/s₅)
0	-1	1	0	0	1	-1	0	0
-1	1	-1	1	1	-1	1	-1	0
1	-1	1	0	0	1	-1	1	-1
0	1	0	0	0	0	1	-1	0
0	1	0	0	0	0	0	0	0
1	-1	1	0	0	0	-1	1	0
-1	1	-1	1	0	-1	1	-1	1
0	-1	1	-1	0	1	-1	1	0
0	0	-1	0	0	0	1	0	0

s₈→s₂ (s₂/s₈)
1	1	0	-1	-1	0	1	0	0
1	1	0	-1	-1	0	0	1	0
0	0	0	0	0	-1	0	0	1
-1	-1	0	1	0	0	-1	0	1
-1	-1	0	0	1	0	0	0	0
0	0	-1	0	0	1	1	0	-1
1	0	0	-1	0	1	1	0	-1
0	1	0	0	0	0	0	0	0
0	0	1	1	0	-1	-1	0	1

s₇→s₃ (s₃/s₇)
-1	0	0	-1	0	1	0	-1	1
0	-1	-1	0	0	0	0	1	0
0	-1	-1	0	0	-1	1	1	0
-1	0	0	-1	-1	1	0	0	1
0	0	0	-1	0	0	0	0	1
1	0	-1	1	0	-1	0	1	0
0	0	1	0	0	0	0	0	0
-1	1	1	0	0	1	0	-1	0
1	0	0	1	1	0	0	0	-1

s₅→s₈ (s₈/s₅)
0	0	-1	1	0	-1	1	0	0
0	-1	1	-1	0	1	-1	1	0
-1	1	-1	0	0	0	1	-1	1
1	-1	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1	0
-1	1	0	0	0	0	1	-1	1
1	-1	1	0	0	1	-1	1	0
0	1	-1	1	1	-1	1	0	0
0	0	1	0	0	1	0	0	0

s₂→s₇ (s₇/s₂)
-1	0	1	0	-1	0	1	1	-1
0	0	0	0	0	0	1	0	0
1	0	-1	0	1	1	-1	0	1
0	0	0	0	1	0	0	0	1
-1	0	1	1	-1	0	1	1	0
0	0	1	0	0	0	1	1	0
1	1	-1	0	1	1	0	0	1
1	0	0	0	1	1	0	0	1
-1	0	1	1	0	0	1	1	0

s₆→s₂ (s₂/s₆)
0	1	0	0	1	0	0	0	-1
1	0	1	1	0	1	0	-1	-1
0	1	1	1	1	0	0	-1	-1
0	1	1	1	1	0	-1	0	-1
1	0	1	1	0	0	0	-1	0
0	1	0	0	0	0	0	0	0
0	0	0	-1	0	0	0	1	0
0	-1	-1	0	-1	0	1	0	1
-1	-1	-1	-1	0	0	0	1	1

s₇→s₄ (s₄/s₇)
1	-1	-1	1	0	-1	0	1	0
-1	0	0	-1	0	0	0	0	1
-1	0	0	-1	-1	1	0	0	1
1	-1	-1	0	0	-1	1	1	0
0	0	-1	0	0	0	0	1	0
-1	0	1	-1	0	1	0	-1	1
0	0	0	1	0	0	0	0	0
1	0	0	1	1	-1	0	1	-1
0	1	1	0	0	1	0	-1	0

s₄→s₅ (s₅/s₄)
0	1	1	0	-1	-1	0	1	0
1	1	1	0	-1	-1	0	0	1
1	1	0	0	0	0	-1	0	1
0	0	0	0	1	0	0	0	0
-1	-1	0	1	0	1	1	0	-1
-1	-1	0	0	1	1	1	0	-1
0	0	-1	0	1	1	0	0	0
1	0	0	0	0	0	0	0	1
0	1	1	0	-1	-1	0	1	1

s₄→s₂ (s₂/s₄)
-1	-1	0	0	1	1	0	0	-1
-1	-1	-1	1	1	1	1	-1	-1
0	-1	0	0	1	1	0	0	-1
0	1	0	0	0	0	0	0	0
1	1	1	0	-1	-1	0	0	1
1	1	1	0	-1	-1	-1	1	1
0	1	0	0	0	-1	0	0	1
0	-1	0	0	0	1	0	0	0
-1	-1	-1	0	1	1	1	0	-1

s₆→s₃ (s₃/s₆)
0	0	1	1	0	0	0	0	-1
0	1	1	1	1	0	0	-1	-1
1	1	1	1	1	1	-1	-1	-1
1	1	1	1	1	0	0	-1	-1
0	1	1	1	0	0	0	0	-1
0	0	1	0	0	0	0	0	0
0	0	-1	0	0	0	0	0	1
0	-1	-1	-1	0	0	0	1	1
-1	-1	-1	-1	-1	0	1	1	1

s₈→s₄ (s₄/s₈)
-1	-1	0	1	1	-1	-1	0	1
-1	-1	0	1	0	0	-1	0	1
0	0	0	-1	0	0	1	0	0
1	1	-1	-1	-1	1	1	1	-1
1	0	0	-1	0	0	1	0	0
-1	0	0	1	0	0	-1	0	1
-1	-1	1	1	1	-1	-1	0	1
0	0	0	1	0	0	0	0	0
1	1	0	-1	0	1	1	0	-1

s₇→s₆ (s₆/s₇)
-1	1	1	-1	-1	1	0	-1	1
1	0	0	0	0	-1	0	1	0
1	0	-1	1	0	-1	0	1	0
-1	0	1	-1	0	1	0	-1	1
-1	0	0	0	0	1	0	0	0
1	-1	-1	1	1	-1	1	1	-1
0	0	0	0	0	1	0	0	0
-1	1	1	-1	0	1	0	-1	1
1	0	0	1	0	-1	0	1	0

s₅→s₇ (s₇/s₅)
0	-1	1	0	0	0	-1	1	0
-1	1	-1	1	0	-1	1	-1	1
1	-1	1	-1	0	1	-1	1	0
0	1	-1	0	0	0	1	0	0
0	0	0	0	0	0	1	0	0
0	-1	1	0	0	1	-1	1	0
-1	1	-1	1	1	-1	1	-1	1
1	-1	1	0	0	1	-1	1	0
0	1	0	0	0	0	1	0	0

s₃→s₈ (s₈/s₃)
-1	1	0	-1	1	0	-1	1	0
1	-1	0	1	-1	0	1	-1	1
0	0	0	0	0	0	0	1	0
-1	1	0	-1	1	0	0	1	0
1	-1	0	1	-1	1	1	-1	1
0	0	0	0	1	0	0	1	0
-1	1	0	0	1	0	0	1	0
1	-1	1	1	-1	1	1	-1	1
0	1	0	0	1	0	0	1	0

s₈→s₉ (s₉/s₈)
-1	0	1	1	0	-1	-1	0	1
0	0	1	1	0	-1	-1	0	1
1	1	0	0	0	0	0	0	0
1	1	0	-1	0	1	1	0	-1
0	0	0	0	0	1	1	0	-1
-1	-1	0	1	1	0	0	0	0
-1	-1	0	1	1	0	-1	0	1
0	0	0	0	0	0	0	0	1
1	1	0	-1	-1	0	1	1	0

s₇→s₉ (s₉/s₇)
-1	0	1	0	0	1	0	-1	0
0	0	0	1	1	0	0	0	-1
1	0	0	1	1	0	0	0	-1
0	1	1	0	0	1	0	-1	0
0	1	1	0	0	0	0	0	0
1	0	0	1	0	-1	0	1	0
0	0	0	0	0	0	0	0	1
-1	0	0	-1	0	1	0	0	1
0	-1	-1	0	0	0	1	1	0

s₆→s₉ (s₉/s₆)
0	-1	-1	0	0	0	0	0	1
-1	-1	-1	-1	0	0	0	1	1
-1	-1	-1	-1	-1	0	1	1	1
0	-1	-1	-1	-1	0	1	1	1
0	0	-1	-1	0	0	0	1	1
0	0	0	0	0	0	0	0	1
0	0	1	1	0	0	0	0	0
0	1	1	1	1	0	0	0	-1
1	1	1	1	1	1	0	-1	-1

s₄→s₉ (s₉/s₄)
-1	-1	0	0	0	1	1	0	-1
-1	-1	-1	0	1	1	1	0	-1
0	-1	-1	0	1	1	0	0	0
0	0	0	0	0	0	0	0	1
0	1	1	0	-1	-1	0	1	1
1	1	1	0	-1	-1	0	1	1
1	1	0	0	0	0	0	0	1
0	0	0	0	1	1	0	0	0
-1	-1	0	1	1	1	1	0	-1

s₂→s₉ (s₉/s₂)
1	0	-1	0	1	0	-1	0	1
0	0	0	0	0	0	0	0	1
-1	0	1	0	-1	0	1	1	0
0	0	0	0	0	0	1	1	0
1	0	-1	0	1	1	0	0	1
0	0	0	0	1	1	0	0	1
-1	0	1	1	0	0	1	1	0
0	0	1	1	0	0	1	1	0
1	1	0	0	1	1	0	0	1

s₂→s₁ (s₁/s₂)
1	1	-1	-1	1	1	-1	-1	1
1	0	0	0	0	0	0	0	0
-1	0	1	1	-1	-1	1	1	-1
-1	0	1	0	0	0	0	0	0
1	0	-1	0	1	1	-1	-1	1
1	0	-1	0	1	0	0	0	0
-1	0	1	0	-1	0	1	1	-1
-1	0	1	0	-1	0	1	0	0
1	0	-1	0	1	0	-1	0	1

s₃→s₁ (s₁/s₃)
0	-1	1	1	-1	0	0	-1	1
-1	1	0	0	1	-1	-1	1	0
1	0	0	0	0	0	0	0	0
1	0	0	0	-1	1	1	-1	0
-1	1	0	-1	1	0	0	1	-1
0	-1	0	1	0	0	0	0	0
0	-1	0	1	0	0	0	-1	1
-1	1	0	-1	1	0	-1	1	0
1	0	0	0	-1	0	1	0	0

s₄→s₁ (s₁/s₄)
-1	-1	0	1	0	0	1	0	-1
-1	-1	0	0	1	1	0	0	-1
0	0	-1	0	1	1	0	-1	0
1	0	0	0	0	0	0	0	0
0	1	1	0	-1	-1	0	1	0
0	1	1	0	-1	-1	0	0	1
1	0	0	0	0	0	-1	0	1
0	0	-1	0	1	0	0	0	0
-1	-1	0	0	0	1	1	0	-1

s₅→s₁ (s₁/s₅)
0	0	-1	1	1	-1	0	0	0
0	-1	1	0	0	1	-1	0	0
-1	1	0	0	0	0	1	-1	0
1	0	0	0	0	0	0	1	-1
1	0	0	0	0	0	0	0	0
-1	1	0	0	0	0	0	-1	1
0	-1	1	0	0	0	-1	1	0
0	0	-1	1	0	-1	1	0	0
0	0	0	-1	0	1	0	0	0

s₆→s₁ (s₁/s₆)
1	0	0	0	0	1	0	-1	0
0	1	0	0	1	0	0	0	-1
0	0	1	1	0	0	0	0	-1
0	0	1	1	0	0	0	-1	0
0	1	0	0	1	0	-1	0	0
1	0	0	0	0	0	0	0	0
0	0	0	0	-1	0	1	0	0
-1	0	0	-1	0	0	0	1	0
0	-1	-1	0	0	0	0	0	1

s₇→s₁ (s₁/s₇)
1	-1	-1	1	0	-1	1	1	-1
-1	0	0	-1	0	1	0	0	0
-1	0	0	-1	0	1	0	-1	1
1	-1	-1	1	0	-1	0	1	0
0	0	0	0	0	-1	0	1	0
-1	1	1	-1	-1	1	0	-1	1
1	0	0	0	0	0	0	0	0
1	0	-1	1	1	-1	0	1	-1
-1	0	1	0	0	1	0	-1	0

s₈→s₁ (s₁/s₈)
1	1	0	-1	-1	0	1	1	-1
1	1	0	-1	-1	0	1	0	0
0	0	0	0	0	0	-1	0	1
-1	-1	0	1	1	-1	-1	0	1
-1	-1	0	1	0	0	0	0	0
0	0	0	-1	0	1	1	0	-1
1	1	-1	-1	0	1	1	0	-1
1	0	0	0	0	0	0	0	0
-1	0	1	1	0	-1	-1	0	1

Matrices by Size

Component Relations

The relations in the first column show how the second row of the matrix table can be derived from the first. The second column shows how the third row can be derived from the second. The list is not exhaustive.

s₉/s₁ - s₈/s₁ = s₁/s₉
s₈/s₁ - s₇/s₁ = s₂/s₉
s₇/s₁ - s₆/s₁ = s₃/s₉
s₆/s₁ - s₅/s₁ = s₄/s₉
s₅/s₁ - s₄/s₁ = s₅/s₉
s₄/s₁ - s₃/s₁ = s₆/s₉
s₃/s₁ - s₂/s₁ = s₇/s₉
s₂/s₁ - 1 = s₈/s₉

s₁/s₉ + s₂/s₉ = s₃/s₈
s₂/s₉ + s₃/s₉ = s₅/s₇
s₃/s₉ + s₄/s₉ = s₇/s₆
s₄/s₉ + s₅/s₉ = s₉/s₅
s₅/s₉ + s₆/s₉ = s₈/s₄
s₆/s₉ + s₇/s₉ = s₆/s₃
s₇/s₉ + s₈/s₉ = s₄/s₂
(s₈/s₉ + 1 = s₂/s₁)

s₄/s₅ + s₁/s₉ = 1
s₄/s₂ + 1 = s₃/s₁)
s₈/s₄ + 1 = s₆/s₂
s₇/s₃ - s₁/s₉ = s₆/s₃
s₂/s₇ + 1 = s₉/s₅
s₈/s₅ + 1 = s₆/s₃)
s₃/s₈ + 1 = s₇/s₄
(s₈/s₉ + 1 = s₂/s₁)

The 9-D Equations

back

0	0	0	0	0	0	0	0	1		1
0	0	0	0	0	0	0	1	1		2
0	0	0	0	0	0	1	1	1		3
0	0	0	0	0	1	1	1	1		4
0	0	0	0	1	1	1	1	1	×	5
0	0	0	1	1	1	1	1	1		6
0	0	1	1	1	1	1	1	1		7
0	1	1	1	1	1	1	1	1		8
1	1	1	1	1	1	1	1	1		9

0	0	0	0	0	0	0	0	1		1
0	0	0	0	0	0	0	1	1		2
0	0	0	0	0	0	1	1	1		3
0	0	0	0	0	1	1	1	1		4
0	0	0	0	1	1	1	1	1	×	5
0	0	0	1	1	1	1	1	1		6
0	0	1	1	1	1	1	1	1		7
0	1	1	1	1	1	1	1	1		8
1	1	1	1	1	1	1	1	1		9

0	0	0	0	0	0	0	0	1		1
0	0	0	0	0	0	0	1	1		2
0	0	0	0	0	0	1	1	1		3
0	0	0	0	0	1	1	1	1		4
0	0	0	0	1	1	1	1	1	×	5
0	0	0	1	1	1	1	1	1		6
0	0	1	1	1	1	1	1	1		7
0	1	1	1	1	1	1	1	1		8
1	1	1	1	1	1	1	1	1		9