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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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