Elementary matrix transformations and Arzashkun: Difference between pages

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[[Image:Arzashburning.PNG|right|thumb|300px|The burning of Arzashkun as depicted on the bronze door coverings of Shalmaneser III from Balawat.]]

[[ja:行列の基本変形]]

'''Arzashkun''' ([[Armenian language|Armenian]]: Արզաշկուն) was located north of [[Lake Van]], double walled and with towers <ref>Journal of the American Oriental Society - Page 360</ref> <ref>Shalmaneser III and the Establishment of the Assyrian Power, by A. T. Olmstead p.360</ref>. The capital of the early [[kingdom of Urartu]] in the [[9th century BC]], before [[Sarduri I]]'s move of the capital to [[Tushpa]] ([[Van, Turkey|Van]]) in [[832 BC]], some 150 km to the south-east.

Arzashkun was captured by [[Shalmaneser III]] in the 840s BC.

== Name ==

'''Elementary matrix transformations''' or '''elementary row and column transformations''' are [[Linear transformation|linear transformations]] which are normally used in [[Gauss elimination method|gauss elimination]] to solve a set of linear equations.

Arzashkun seems to be the [[Assyrian]] form of an [[Urartian language|Urartian]] name ending in -ka formed from a proper name Arzash, which recalls the name Arsene, Arsissa, applied by the ancients to part of [[Lake Van]]. Arzashkun might represent the Ardzik of the [[Armenians|Armenian]] historians, west of [[Malazgirt|Malasgert]]. <ref>History of Egypt - Page 91 by G. Maspero</ref>

== Background ==

We distinguish three types of elementary transformations and their corresponding matrices:

Arzashkun was hidden, and protected against attack, by an extent of dense forest almost impassable to a regular army. <ref>Passing of the Empires 850 BC to 330 BC</ref>

# '''Row-switching''' transformations,

# '''Row-multiplying''' transformations,

# '''Linear combinator''' transformations.

== Location ==

===1. Row-switching transformations===

There is no agreement as to the precise ___location of Arzashkun. It may correspond to the ''Ardzik'' of the Armenian historians, situated west of [[Malasgert]], at the [[Arzanias]] river, roughly {{coor dm|39|10|N|42|10|E|}}, in what is now the Turkish [[Muş Province]].

This transformation, ''T_ij'', switches all matrix elements on row i with their counterparts on row j. The matrix resulting in this transformation is:<br>

:<math>

T_{i,j} = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 0 & & 1 & & \\ & & & \ddots & & & & \\ & & 1 & & 0 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix},\quad </math>

:That is, ''T_ij'' is the matrix produced by exchanging row ''i'' and row ''j'' of the identity matrix.

====Properties====

:*The matrix ''T_ij'' is [[Square_matrix#Square_matrices_and_related_definitions|square]].

:*The inverse of this matrix is itself: ''T_ij^-1=T_ij''.

:*Since the [[Determinant|determinant]] of the [[Identity_matrix|identity matrix]] is unity, ''det[T_ij]=-1''. It follows that for any conformable square matrix ''A'': ''det[T_ijA]=-det[A]''.

This city is located variously by different scholars in the region of [[Lake Urmia]], in that of Lake Van at Malazgirt or at Bostankaya between Malagirt and Patnos or lying west or north of [[Lake Van]]. <ref>The Cambridge Ancient History - Page 335 by John Boardman</ref>

===2. Row-multiplying transformations===

This transformation, ''T_i(m)'', multiplies all elements on row i with ''m''. The matrix resulting in this transformation is:<br>

:<math>

T_i(m) = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 1 & & & & & \\ & & & m & & & & \\ & & & & & 1 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix},\quad </math>

== Fall of Arzashkun ==

====Properties====

At the headwaters of the river [[Tigris]], there appears in the ninth century, B.C., an organized state of [[Urartu]]. Shalmaneser regarded it as so menacing to [[Assyria]]'s interest that he undertook an expedition in 857, claimed to destroyed the capital Arzashkun <ref>The Ancient Assyrians - Page 12

:*The inverse of this matrix is: ''T_i(m)^-1=T_i(1/m)''.

by Mark Healy</ref>, penetrated as far as [[Lake Van]], and left his inscription on [[Mount Irritia]]. <ref>Aram and Israel - Page 105 by Emil Gottlieb Heinrich Kraeling</ref>

:*The matrix and its inverse are [[Diagonal_matrix|diagonal matrices]].

:*''det[T_i(m)]=m''. Therefore for a conformable square matrix ''A'': ''det[T_i(m)A]=m det[A]''.

Shalmaneser on his ''Black Obelisk'' records this campaign:

===3. Linear combinator transformations===

This transformation, ''T_ij(m)'', subtracts row i multiplied by ''m'' from row j. The matrix resulting in this transformation is:<br>

:<math>

T_{i,j}(m) = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 1 & & & & & \\ & & & \ddots & & & & \\ & & -m & & 1 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix}

</math>

:(35-44) In the third year of my reign, Ahuni, son of Adini, was frightened before my mighty weapons and retreated from Til-barzip, his royal city. I crossed the Euphrates. I seized for myself the city of Ana-Assur-utir-asbat, which lies on the other side of the Euphrates, on the Sagur river, which the Hittite people called Pitru. When I returned, I entered the passes of the land of Alzi; the lands of Alzi, Suhni, Daiaeni, Tumme, Arzashkunu, the royal city of [[Aramu|Arame]], the Armenian (king), Gilzânu, and Hubushkia (I conquered).[http://www.kchanson.com/ANCDOCS/meso/obelisk.html]

====Properties====

:*''T_ij(m)^-1=T_ij(-m)'' (inverse matrix).

:*The matrix and its inverse are [[Triangular_matrix|triangular matrices]].

:*''det[T_ij(m)]=1''. Therefore, for a conformable square matrix ''A'': ''det[T_ij(m)A]=det[A]''.

==See alsoReferences ==

{{Reflist}}

== See also ==

*[[Urartu]]

*[[Sugunia]]

[[Category:Urartu]]

*[[Linear algebra]]

[[Category:History of Anatolia]]

*[[Gauss elimination method]]

[[Category:History of Armenia]]

[[hy:Արզաշկուն]]