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 Matrix Representation of Quaternions
 - Private Edition

 Copyright MIYAMA. 2025 Nov

 kazutomimiyama.sakura.ne.jp
 KazutomiMiyamaSub@gmail.com

 Thanks: DeepSeek
 Microsoft, Copilot

 google transration Service

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<Introduction> 

 This is an entry for a concept included
 in a writing archive called "Three-
Dimensional Essays" (available on the 
Vector website) that I wrote about 15 
years ago. However, the archive was 
mostly devoted to explaining three-
dimensional octahedral matrices (48 
elements. See something like 48 moves 
or 48 idols. Buddhist auspicious number.), 
and this concept was only briefly 
touched upon. (The attempted proof was 
about 30 years ago.) This time, I have 
extracted that section here and provided 
an explanation. Of course, in the 
rigorous theories of my predecessors, 
the concept is given a rather 
intimidating name (apparently a Lie 
group), but since I am inferring based 
on a simple process, I will prioritize 
using the temporary, convenient name 
that was used in that process. Since 
I came up with the name myself, I am not 
particularly hoping that it will become 
established, but since it is a name that 
is paired with the concept, it may be 
fun to read. Of course, that is assuming 
you can grasp the concept explained. 

---------------------------------------- 
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<A simple explanation of quaternions> 

 The quaternions discovered by Hamilton 
are a type of imaginary triplet, and 
they form a group under multiplication, 
but the commutative law does not hold 
when they are multiplied. However, if 
the sign is reversed, the law does hold. 
-The product of triplet units has a 
rotational conceptual parity, and 
reversing the rotation reverses the sign 
of the value. (Conceptually homologous 
to the line above) In other words, 
(where ^ indicates factorial)

 i^2 = j^2 = k^2 = -1

 i*j = -j*i = -k
 j*k = -k*j = -i
 k*i = -i*k = -j

 i*j*k = -1
 k*j*i = +1

 holds. 

----------------------------------------- 
<The matrix that started it all>

 By chance, I discovered that when the 
matrix

 (0 -1)
 (1  0)

 is squared, it becomes the negative of 
the identity element, (-1 0) (0 -1) = -E. 
(The details of how this came about are 
at the end of the document.) By regarding 
this as congruent with the imaginary unit 
i, I attempted to ambitiously define a 
matrix equivalent to a quaternion by 
combining matrix operations, and this 
document is a record of my trial and 
error. To state the conclusion first, 
this could not be achieved with a 2*2 
square matrix, so a 4*4 matrix was 
needed as an extended representation of 
the definition, and I was able to 
achieve the desired result. 

---------------------------------------- 
<The history of trial and error>

 Naturally, I had no clues, so I 
wondered if I could define a quaternion 
with a 2*2 matrix. Although this attempt 
alone failed, by conceptually expanding 
the failed equation we were able to 
obtain the final 4x4 matrix. First, we 
define the following: (0 is zero)

      (1  0)     (0 -1)
 Ed = (0 -1) D = (1  0)

 D means different 
(Ed is the deferent of the identity 
element)

 We will also write the results first, 
but we set another matrix as

     (0 1)
 S = (1 0) (S means same)

 and tried to see if these three would 
form a product group. This alone did not 
work, but we found that

 (iEd) (D) (iS)

 satisfy the conditions for a quaternion. 
Below is the proof (i is the imaginary 
unit)... (*1)

 (1 0) (0-1)   (0 -1)
 (0-1) (1 0) = (-1 0) 
----------------------------------
 Ed * D = -S Therefore 
---------------------------------- 
(iEd) * D = -(iS)
 > equivalent to a quaternion

 (0-1) (0 1)   (-1 0)
 (1 0) (1 0) = (0  1) 
----------------------------------
 D * S = -Ed Therefore 
----------------------------------
 D * (iS) = -(iEd)
 > equivalent to a quaternion

 (0 1) (1 0)   (0-1)
 (1 0) (0-1) = (1 0) 
---------------------------------
 S * Ed = + D Therefore 
---------------------------------
 (iS) * (iEd) = - D
 >This is also equivalent to a 
quaternion 

---------------------------------------- 
<Matrix representation without using the 
 imaginary unit i>

 We were able to obtain a product group 
equivalent to a quaternion, but this was 
obtained by forcibly adding an existing 
imaginary unit as a borrowed entity, 
so this does not represent a quaternion 
in matrix representation. Here, we will 
use the property that

      (0 -1)
 i ≡ (1  0) = D.

 The method of "expanding" a 2*2 matrix 
to a 4*4 matrix can be thought of as:

       (a b)
 A22 = (c d)

       (a*E b*E)
 A44 = (c*E d*E)

 (E is the identity element for
 multiplication of a 2*2 square matrix)
 Of course,

 A22 ≡ A44 Here,
 i*A22 ≡ D*A44,

 and as D and E are both 2*2 square 
matrices, we can think of D as a 
coefficient of a 4*4 matrix 
(Copilot described this as an expansion 
to a tensor. I don't know the details.) 
Therefore, we consider

 i*Ed22 ≡ D*Ed44 (*2)
 i*S22  ≡ D*S44  (*3).

 Therefore, by expanding the matrix, we 
obtain a 4 * 4 matrix.

 *O: O is the 2 * 2 zero matrix.
 *E is the 2 * 2 multiplication identity.

        (E  O)
 Ed44 = (O -E)

       (O -E)
 D44 = (E  O) (*4)

       (O E)
 S44 = (E O)

 Consider this. Treating D as the 
imaginary unit in matrices as a 
coefficient, we obtain the combination 
equivalent to equation *1.

 (D  O) (O -E) (O D)
 (O -D) (E  O) (D O)

 As a result of operational verification, 
we were able to confirm that 
multiplicationbehaves the same as 
quaternions.

(The proof is omitted because it is 
homologous to *1; the proof is at the 
end of the file.)

 I mentioned *4 earlier, but it can be 
derived from the product of *2 and *3.

---------------------------------------- 
<Another combination>

 Interestingly, we have found another 
set of matrices equivalent to 
quaternions. In the previous section, we 
extended and substituted the imaginary 
behavior matrix D in place of i, but 
this time we will consider the imaginary 
behavior in terms of a 4*4 skeleton. In 
other words, in the previous section, 
the extension was

 iEd≡D*Ed44
 iS ≡D*S44,

 but we can now think of it as

 iEd≡Ed*D44
 iS ≡ S*D44.

 This means that we cannot use D44 in 
the position of D in 2*2 (it will break 
down when we do multiplication). Here, 
(E and E44 are identity elements for 
multiplication)

 D44 = E*D44 

(an element of the combination from the 
previous section), so we can similarly 
reverse the positions of the 
coefficients and skeleton and think of 
it as D≡D*E44. When this is combined 
with the new extension 4*4 of iEd and 
iS, this also matches the product group 
as a quaternion. 
---------------------------------------- 
<Product proof of Ed*D44 D*E44 S*D44>

 Ed*D=-S Therefore

 (O -Ed)   (D O)   (O  S)      (O -S)
 (Ed  O) * (O D) = (-S O) = (-)(S  O)

 EdD44 * DE44 = −SD44

 D*S=-Ed Therefore

 (D O)   (O -S)   (O  Ed)      (O -Ed)
 (O D) * (S  O) = (-Ed O) = (-)(Ed  O)

 DE44 * SD44 = −EdD44

 S*Ed=+D Therefore

 (O -S)   (O -Ed)   (-D O)      (D O)
 (S  O) * (Ed  O) = (O -D) = (-)(O D)

 SD44 * EdD44 = -DE44

 This concludes the illustration of the 
important concepts. 

---------------------------------------- 
<Rigorous Verification>

 I am not well-informed as to whether 
this is called a tensor operation, and I 
am not sure whether this alone will be 
enough to prove it, so as a check I will 
write down the verification in 
elementary expression of 4 x 4. 
Parentheses, zeros, and zero matrices 
have been omitted to avoid complicating 
things. The vertical and horizontal 
lines in the middle are guide lines.

---------------------------------------- 
<'Gold' i tendency from D22>

・Squared Check

    -1|              -1|           -1    |
 1    |           1    |               -1|
-------------  * -------------  =  -------------
      |     1          |     1           |-1
      |-1              |-1               |    -1

 (D  O)^2
 (O -D) = -E44

      |-1               |-1         -1    |
      |    -1           |    -1         -1|
-------------  *  -------------  =  -------------
 1    |            1    |                 |-1
     1|                1|                 |    -1

 (O -E)^2
 (E  O) = -E44

      |    -1           |    -1     -1    |
      | 1               | 1             -1|
-------------  *  -------------  =  -------------
    -1|               -1|                 |-1
 1    |            1    |                 |    -1

 (O D)^2
 (D O) = -E44 

・Confirmation of cumulative group behavior

    -1|                 |-1             |     1            |    -1
 1    |                 |    -1         |-1                | 1
-------------  *  ------------- = ------------- = (-)-------------
      |     1      1    |              1|                -1|
      |-1              1|         -1    |             1    |

 (D  O)   (O -E)   (O -D)      (O D)
 (O -D) * (E  O) = (-D O) = (-)(D O)

      |-1               |    -1        1|               -1|
      |    -1           | 1       -1    |            1    |
-------------  *  ------------- = ------------- = (-)------------
 1    |               -1|               |    -1           |     1
     1|             1   |               | 1               |-1

 (O -E)   (O D)   (-D O)       (D  O)
 (E  O) * (D O) = (O  D) = (-) (O -D)

      |    -1         -1|               | 1                |-1
      | 1          1    |               |     1            |    -1
-------------  *  ------------- = -------------- = (-)------------
    -1|                 |     1   -1    |             1    |
 1    |                 |-1           -1|                 1|

 (O D)   (D  O)   (O  E)       (O -E)
 (D O) * (O -D) = (-E O) = (-) (E  O)

---------------------------------------- 
<'Plum' i tendency from D44>

・Squared Check

      |-1               |-1         -1    |
      |    1            |    1          -1|
------------  *  -------------  =  --------------
 1    |           1     |                 |-1
    -1|               -1|                 |    -1

 (O -Ed)^2
 (Ed  O) = -E44

    -1|               -1|           -1    |
 1    |            1    |               -1|
-------------  *  -------------  =  -------------
      |    -1           |    -1           |-1
      | 1               | 1               |    -1

 (D O)^2
 (O D) = -E44

      |    -1           |    -1     -1    |
      |-1               |-1             -1|
-------------  *  -------------  =  -------------
     1|                1|                 |-1
 1    |            1    |                 |    -1

 (O -S)^2
 (S  O) = -E44

---------------------------------------- 
 ・Confirmation of cumulative group 
behavior

      |-1             -1|               |     1            |    -1
      |     1      1    |               | 1                |-1
-------------  *  ------------- = ------------- = (-)-------------
 1    |                 |    -1       -1|                 1|
    -1|                 | 1       -1    |             1    |

 (O -Ed)   (D O)   (O  S)       (O -S)
 (Ed  O) * (O D) = (-S O) = (-) (S  O)

    -1|                 |    -1         | 1                |-1
 1    |                 |-1             |    -1            |     1
-------------  *  ------------- = ------------- = (-)-------------
      |    -1          1|         -1    |             1    |
      | 1          1    |              1|                -1|

 (D O)   (O -S)   (O  Ed)       (O -Ed)
 (O D) * (S  O) = (-Ed O) = (-) (Ed  O)

      |    -1            |-1            1|               -1|
      |-1                |     1   -1    |            1    |
-------------  *   ------------- = ------------- = (-)------------
     1|             1    |               |     1           |    -1
 1    |                -1|               |-1               | 1

 (O -S)   (O -Ed)   (-D O)       (D O)
 (S  O) * (Ed  O) = (O -D) = (-) (O D)

---------------------------------------- 
<What's the point of just using compound 
 notation?>

 As a joke, I tried giving nicknames to 
4*4 matrices. This was an idea I came up 
with 30 years ago, so I have to chuckle 
at how poorly it felt at the time. I got 
two pairs, so I wanted contrasting names, 
and I was also a little skeptical about 
the Western leaning of general 
mathematical symbols, so I decided to 
use Chinese characters. 金(kim,Gold) and 
李(Li,Plum) are common surnames in China 
and Korea. Both have auspicious meanings.
 One is a mineral, the other a lush tree. 
I came up with this pair by considering 
whether to use a compact 2*2 imaginary 
behavior matrix or one that uses the full 
4*4 space, but it seems my old self ended 
up adopting this by stretching the idea of 
​​compact being a mineral and lush space 
being a tree. Let's summarize the 
definitions again here
---------------------------------------- 

                            (D   O)
 金Ed = GoldEd = D * Ed44 = (O - D)

                             (O -E)
 金(D) = Gold(D) = E * D44 = (E  O)

                           (O S)
 金S = GoldS   = D * S44 = (S O) 

---------------------------------------- 

                            (O -Ed)
 李Ed = PlumEd = Ed * D44 = (Ed  O)

                             (D O)
 李(D) = Plum(D) = D * E44 = (O D)

                           (O -S)
 李S = PlumS   = S * D44 = (S  0) 

---------------------------------------- 
*If we strictly define here that

 "Gold is something whose coefficient
 matrix is ​​D"

 and

 "Plum is something whose skeleton
 matrix has the properties of D",

 then Gold(D) and li(D) would be written 
in reverse. However, if these 
multiplication groups are closed and do 
not interfere, it would be difficult to 
understand if names written with 
different Chinese characters were mixed 
in with the groups, so I left it as it 
is, as it is also a convenient name. 
Alternatively, it might be better to 
write Gold(D*E44) li(E*D44), but if the 
only reason for the names is to make a 
distinction, this would also be a 
complicated format. 

---------------------------------------- 
<Development - Commutative Groups>

 According to my memory of trials 30 
years ago, when these two groups of six 
matrices are multiplied and "outcrossed," 
nine new matrices are obtained, which 
respectively constitute three non-
interfering multiplicative commutative 
groups. Explaining this would be too 
complicated, so I will omit it.
 However, as a generated product, there 
is the following relationship between 
them. This is described below. The 
commutative group matrices obtained by 
combining the intersections of the 
horizontal Gold and the vertical Plum 
are the respective commutative group 
matrices. Here, I will simply write "Cmt"
 
      Gold Gold Gold
 Plum Cmt  Cmt  Cmt
 Plum Cmt  Cmt  Cmt
 Plum Cmt  Cmt  Cmt

 If this were to be redesigned 
artistically, it would look like

 E    Gold Gold Gold E
 Plum Cmt  Cmt  Cmt  Plum
 Plum Cmt  Cmt  Cmt  Plum
 Plum Cmt  Cmt  Cmt  Plum
 E    Gold Gold Gold E

 It looks exactly like the mandala of 
the Womb Realm. Buddhist auspicious 
numbers and Buddhist art reflect a sense 
of Indian mathematics. 
:I have heard that group theory can also 
be seen in Islamic geometric designs 
(New Mathematics Study Method, Kodansha 
Bluebacks by Key Tohyama) 

---------------------------------------- 
<Appendix: How i≡D was discovered>

 Without finding this, the extension to 
quaternion behavior would not have been 
possible. It was discovered by chance.
 Consider an arbitrary point on a circle 
of radius 1. Let the angle at that point 
from the reference line be theta. The 
matrix for rotational transformation on 
the xy plane is as follows (this can be 
proven using elementary geometry):

 cos(theta) -sin(theta)
 sin(theta)  cos(theta)

 This can be transformed into the 
following using matrix addition:

            (1 0)            (0 -1)
 cos(theta)*(0 1)+sin(theta)*(1  0)

            : E              : D

 This was important. I realized that 
this seemed to be homologous to de 
Moivre's formula, which indicates a 
point on a unit circle of the same 
radius 1 on the complex plane, rather 
than the xy plane, and since in

 cos(theta)+i*sin(theta),

 i^2 = -1, if we also square D, the 
concept equivalent to -1 in a matrix is 
​​the negative value of the integral 
identity, -E. Doing some simple 
calculations, I found that

       (0 -1)   (0 -1)   (-1 0)
 D^2 = (1  0) * (1  0) = (0 -1) = -E

 I was pinched by the nose. It felt like 
discovery was a painful experience. (30 
years ago) It is one proverb in Japan.
'fox pinched your nose' when someone 
feels sence of wonder. :)

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