The Hurwitz Theorem on Sums of Squares by Linear Algebra

نویسنده

  • KEITH CONRAD
چکیده

(1.1) (x1 + x 2 2)(y 2 1 + y 2 2) = (x1y1 − x2y2) + (x1y2 + x2y1). There is also an identity like this for a sum of four squares: (x1 + x 2 2 + x 2 3 + x 2 4)(y 2 1 + y 2 2 + y 2 3 + y 2 4) = (x1y1 − x2y2 − x3y3 − x4y4) + (1.2) (x1y2 + x2y1 + x3y4 − x4y3) + (x1y3 + x3y1 − x2y4 + x4y2) + (x1y4 + x4y1 + x2y3 − x3y2). This was discovered by Euler in the 18th century, forgotten, and then rediscovered in the 19th century by Hamilton in his work on quaternions. Shortly after Hamilton’s rediscovery of (1.2) Cayley discovered a similar 8-square identity. In all of these sum-of-squares identities, the terms being squared on the right side are all bilinear expressions in the x’s and y’s: each such expression, like x1y2 +x2y1 for sums of two squares, is a linear combination of the x’s when the y’s are fixed and a linear combination of the y’s when the x’s are fixed. It was natural for mathematicians to search for a similar 16-square identity next, but they were unsuccessful. At the end of the 19th century Hurwitz [3] proved his famous “1,2,4,8 theorem,” which says that further identities of this kind are impossible.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

The Hurwitz Theorem on Sums of Squares by Representation Theory

This was discovered by Euler in the 18th century, forgotten, and then rediscovered in the 19th century by Hamilton in his work on quaternions. Shortly after Hamilton’s rediscovery of (1.2) Cayley discovered a similar 8-square identity. In all of these sum-of-squares identities, the terms being squared on the right side are all bilinear expressions in the x’s and y’s: each such expression, like ...

متن کامل

The Hurwitz Theorem on Sums of Squares

This was discovered by Euler in the 18th century, forgotten, and then rediscovered in the 19th century by Hamilton in his work on quaternions. Shortly after Hamilton’s rediscovery of (1.2) Cayley discovered a similar 8-square identity. In all of these sum-of-squares identities, the terms being squared on the right side are all bilinear expressions in the x’s and y’s: each such expression, like ...

متن کامل

The Almost Sure Convergence for Weighted Sums of Linear Negatively Dependent Random Variables

In this paper, we generalize a theorem of Shao [12] by assuming that is a sequence of linear negatively dependent random variables. Also, we extend some theorems of Chao [6] and Thrum [14]. It is shown by an elementary method that for linear negatively dependent identically random variables with finite -th absolute moment the weighted sums converge to zero as where and is an array of...

متن کامل

A Generalization of the Doubling Construction for Sums of Squares Identities

The doubling construction is a fast and important way to generate new solutions to the Hurwitz problem on sums of squares identities from any known ones. In this short note, we generalize the doubling construction and obtain from any given admissible triple [r, s, n] a series of new ones [r+ ρ(2m−1), 2s, 2n] for all positive integer m, where ρ is the Hurwitz–Radon function.

متن کامل

Distributive lattices with strong endomorphism kernel property as direct sums

Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem  2.8}). We shall determine the structure of special elements (which are introduced after  Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2008