Perfect Squares and Quadratic Forms

Abstract

In this note we define the inner product of two vectors by a new form. By this way, we show
that every perfect square is a quadratic form of Gram matrix of coefficients of related linear combination.
Moreover, we give a different proof that determinant of the quadratic matrix of a perfect square equals to
zero. We can obtain an equivalence relation between the quadratic matrices of the same perfect square. It
means that our method gives a new aspect of quadratic forms and Pythagorean triples are very useful in the
obtaining of the equivalent quadratic matrices of a perfect square.