As an important component of one of the branches of 20th century mathematics called functional analysis, the theory of quadratic forms was first developed by the German mathematician, David Hilbert (1862-1943) during his lifelong tenure as a professor of mathematics at the University of Göttingen in Central Germany. This work was developed circa 1904 to 1906 as a consequence of his interest in the subject of integral equations after hearing a lecture by Erik Holmgren (1872-1943) on the work of Fredholm (1866-1927). He proved that any quadratic form can be brought to the diagonal form by rotation of the axes. His key result is the generalization to quadratic forms in infinitely many variables of the more familiar principal axis theorem of analytical geometry. He demonstrated by mathematical arguments that there exist orthogonal transformations such that in the new variables, the quadratic forms are reduced to sums of squares of the new variables. For the case of two variables ��₁ and ��₂ the quadratic form ��(��₁,₂)=(��₁+��₂)˛=��₁₁��₁˛+��₁₂��₁��₂+��₂₁��₁��₂+��₂₂��₂˛ is reduced to the diagonal form ��₁��₁˛+��₂��₂˛. On the other hand, if ��₁ and ��₂ are the components of a vector �� then the quadratic form �� is given as the matrix product of the transpose of ��, a real symmetric coefficient matrix �� and ��: ��(��₁, ��₂)=��†����.

Nonetheless, if ��₁ and ��₂ are the absolute values of two vectors ��₁ and ��₂ then the square of their difference is given by (|��₁|-|��₂|)˛= ��₁₁|��₁|˛-��₁₂|��₁||��₂|-��₂₁|��₁||��₂|+��₂₂|��₂|˛ where �� becomes the second order singular symmetric Hadamard matrix. The quadratic form for the 3rd order ��(3)=(��₁-��₂+��₃)˛, likewise, ��(4)=(��₁-��₂+��₃-��₄)˛, ��(5)= (��₁-��₂+��₃-��₄+��₅)˛, ��(6)= (��₁-��₂+��₃-��₄+��₅-��₆)˛, ��(7)= (��₁-��₂+��₃-��₄+��₅-��₆+��₇)˛, and ��(8 )= (��₁-��₂+��₃-��₄+��₅-��₆+��₇-��₈)˛ as can be generalized to infinite order of the singular symmetric Hadamard matrices. It can be noted that for the second order Hadamard matrix, ��(2)=(|��₁|-|��₂|)˛= ��₁₁|��₁|˛-��₁₂|��₁||��₂|-��₂₁|��₁||��₂|+��₂₂|��₂|˛ is equivalent to the law of cosine given as (|��₁|-|��₂|)˛= ��₁₁|��₁|˛-2(��₁₂|��₁||��₂|+��₂₁|��₁||��₂|)cos(��)+��₂₂|��₂|˛ and for ��=90° of the direction cosine, the quadratic form is simply reduced into the Pythagorean theorem.