Perfect Squares and Quadratic Forms


Keywords:
Perfect Squares, Quadratic Forms, DeterminantAbstract
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.
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References
A. Auel, O. Biesel, J. Voight, (2023), Stickelberger’s Discriminant Theorem for Algebras, The American Mathematical Monthly, 130:7, 656-670.
K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, second edition, Graduate Text in Mathematics, vol. 84, Springer-Verlag, New York, 1990.
O. T. O’ meara, Introduction to Quadratic Forms, Classics in Mathematics, Springer-Verlag, Berlin , 2000.