Magic Squares as Vectors to a field of Reals

Pattern 1: Multiplication of a scalar into the quantity and the result is of the same type as the original quantity.

displaystyle{ M = left(begin{array}{cccc} 16 & 3 & 2 & 13\ 5 & 10 & 11 & 8\ 9 & 6 & 7 & 12\ 4 & 15 & 14 & 1 end{array}right)} 

is a magic square.

displaystyle{ 3 M = left(begin{array}{cccc} 48 & 9 & 6 & 39\ 15 & 30 & 33 & 24\ 27 & 18 & 21 & 36\ 12 & 45 & 42 & 3 end{array}right)}  

is also a magic square.Pattern 2: Add 2 of the quantity and the result is of the same type as the original quantities.

displaystyle{ left(begin{array}{cccc} 16 & 3 & 2 & 13\ 5 & 10 & 11 & 8\ 9 & 6 & 7 & 12\ 4 & 15 & 14 & 1 end{array}right) + left(begin{array}{cccc} 20 & 16 & 6 & 7\ 2 & 11 & 13 & 23\ 19 & 21 & 4 & 5\ 8 & 1 & 26 & 14 end{array}right) = left(begin{array}{cccc} 36 & 19 & 8 & 20\ 7 & 21 & 24 & 31\ 28 & 27 & 11 & 17\ 12 & 16 & 40 & 15 end{array}right)}  

In the above statement, the sum is also a magic square.Any quantities that obey Pattern one and 2 are vectors, and hence unit vectors can be defined. The norm a.k.a the magnitude can be defined as the constant sum of magic square.

displaystyle{ hat{a} = left(begin{array}{cccc} 1 & 0 & 0 & 0\ 0 & 0 & 0 & 1\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0 end{array}right)  hat{b} = left(begin{array}{cccc} 0 & 1 & 0 & 0\ 0 & 0 & 0 & 1\ 0 & 0 & 1 & 0\ 1 & 0 & 0 & 0 end{array}right)  hat{c} = left(begin{array}{cccc} 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 1 & 0 & 0 & 0\ 0 & 0 & 0 & 1 end{array}right)  hat{d} = left(begin{array}{cccc} 0 & 0 & 0 & 1\ 0 & 1 & 0 & 0\ 1 & 0 & 0 & 0\ 0 & 0 & 1 & 0 end{array}right) } 

 

displaystyle{ hat{e} = left(begin{array}{cccc} 0 & 0 & 0 & 1\ 1 & 0 & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 1 & 0 & 0 end{array}right)  hat{f} = left(begin{array}{cccc} 0 & 0 & 1 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 0 & 1\ 1 & 0 & 0 & 0 end{array}right)  hat{g} = left(begin{array}{cccc} 1 & 0 & 0 & 0\ 0 & 0 & 0 & 1\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0 end{array}right)  hat{h} = left(begin{array}{cccc} 1 & 0 & 0 & 0\ 0 & 0 & 0 & 1\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0 end{array}right) }  

Since there are seven unit vectors we are dealing with a 7th dimensional space.

displaystyle{ 6 hat{a} + 2 hat{b} + hat{c} + 8 hat{d} + 5 hat{e} + 2 hat{f} + 10 hat{g} = left(begin{array}{cccc} 16 & 3 & 2 & 13\ 5 & 10 & 11 & 8\ 9 & 6 & 7 & 12\ 4 & 15 & 14 & 1 end{array}right) } 

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