1.3.6 Show that the multiplicative property of determinants gives the real four- square identity (af +b} +c}+d})(až+b3+c3+dž) = (aja2 – b¡b2 – c1C2 – dịd2)² + (a¡b2+bja2+c¡d2 – dịc2)² + (ajc2 – bịd2 +cja2+d¡b2)² + (ajd2+bịc2 –c¡b2+dja2)². This identity was discovered by Euler in 1748, nearly 100 years before the dis- covery of quaternions! Like Diophantus, he was interested in the case of integer squares, in which case the identity says that (a sum of four squares) × (a sum of four squares) = (a sum of four squares). This was the first step toward proving the theorem that every positive integer is the sum of four integer squares. The proof was completed by Lagrange in 1770.

1.3.6 Show that the multiplicative property of determinants gives the real four- square identity (af +b} +c}+d})(až+b3+c3+dž) = (aja2 – b¡b2 – c1C2 – dịd2)² + (a¡b2+bja2+c¡d2 – dịc2)² + (ajc2 – bịd2 +cja2+d¡b2)² + (ajd2+bịc2 –c¡b2+dja2)². This identity was discovered by Euler in 1748, nearly 100 years before the dis- covery of quaternions! Like Diophantus, he was interested in the case of integer squares, in which case the identity says that (a sum of four squares) × (a sum of four squares) = (a sum of four squares). This was the first step toward proving the theorem that every positive integer is the sum of four integer squares. The proof was completed by Lagrange in 1770.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.2: Determinants

Problem 20EQ

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