Dot Products and Matrix Properties of 4×4 Strongly Magic Squares

Magic squares have been known in India from very early times. The renowned mathematician Ramanujan had immense contributions in the field of Magic Squares. A magic square is a square array of numbers where the rows, columns, diagonals and co-diagonals add up to the same number. The paper discuss about a well-known class of magic squares; the strongly magic square. The strongly magic square is a magic square with a stronger property that the sum of the entries of the sub-squares taken without any gaps between the rows or columns is also the magic constant. In this paper a generic definition for Strongly Magic Squares is given. The matrix properties of 4×4 strongly magic squares dot products and different properties of eigen values and eigen vectors are discussed in detail.


Introduction
Magic squares date back in the first millennium B. C. E in China [1], developed in India and Islamic World in the first millennium C. E, and found its way to Europe in the later Middle Ages [2] and to sub-Saharan Africa not much after [3]. Magic squares generally fall into the realm of recreational mathematics [4,5], however a few times in the past century and more recently, they have become the interest of more-serious mathematicians. Srinivasa Ramanujan had contributed a lot in the field of magic squares. Ramanujan's work on magic squares is presented in detail in Ramanujan's Notebooks [6]. A normal magic square is a square array of consecutive numbers from 1 … where the rows, columns, diagonals and co-diagonals add up to the same number. The constant sum is called magic constant or magic number. Along with the conditions of normal magic squares, strongly magic square have a stronger property that the sum of the entries of the sub-squares taken without any gaps between the rows or columns is also the magic constant [7]. There are many recreational aspects of strongly magic squares. But, apart from the usual recreational aspects, it is found that these strongly magic squares possess advanced mathematical properties.

Magic Square
A magic square of order n over a field where denotes the set of all real numbers is an n th order matrix [ ] with entries in such that Equation (1) represents the row sum, equation (2) represents the column sum, equation (3) represents the diagonal and co-diagonal sum and symbol represents the magic constant. [8]

Magic Constant
The constant in the above definition is known as the magic constant or magic number. The magic constant of the magic square A is denoted as ( ).

Strongly Magic Square (SMS): Generic Definition
A strongly magic square over a field is a matrix [ ] of order × with entries in such that Equation (4) represents the row sum, equation (5) represents the column sum, equation (6) represents the diagonal & co-diagonal sum, equation (7) represents the × sub-square sum with no gaps in between the elements of rows or columns and is denoted as / !0 ( ) / !1 ( ) and is the magic constant.
Note: The 23 order sub square sum with 4 column gaps or 4 row gaps is generally denoted as / 0 ( ) or / 1 ( ) respectively.

Proof
The general form of a 4x4 SMS is given by The characteristic polynomial of A is given by | − g h| = 0 i.e, Simplifying (10)  Thus sum of the eigen values of a SMS is the magic constant .

3.2.3.
(1,1,1, … .1) n is the eigen vector corresponding to the eigen value of a strongly magic square .
Proof The eigen vector X of a matrix A with eigen value λ is given by AX = λX.
By using the fact that the one of the eigen value is and the row sum is also ; we have (1,1,1, … .1) n as the eigen vector corresponding to eigen value . Clarifies the proof have to illustration in the following form The particular 4x4 SMS Sri Rama Chakra is given by  It can be verified that | | = 0.

3.3.2.
The rank of a 4 × 4 strongly magic square is always 3. Proof The general form of a 4x4 SMS is given by It can be verified using matlab which is the rank given above SMS is 3.

3.3.3.
The rank of a 4x4 SMS and