Looking for any complex number solutions to the system of equations: <mtable columnalign="right

Kazeljkaml5n9y

Kazeljkaml5n9y

Answered question

2022-05-15

Looking for any complex number solutions to the system of equations:
| a | 2 + | b | 2 + | c | 2 = 1 3 a ¯ b + a c ¯ + b ¯ c = 1 6 ( 2 + 3 i ) .

Answer & Explanation

partyjnopp9wa

partyjnopp9wa

Beginner2022-05-16Added 17 answers

By Cauchy-Schwarz, we must have
( | a | 2 + | b | 2 + | c | 2 ) ( | c ¯ | 2 + | a ¯ | 2 + | b ¯ | 2 ) | a c ¯ + b a ¯ + c b ¯ | 2
( | a | 2 + | b | 2 + | c | 2 ) ( | c ¯ | 2 + | a ¯ | 2 + | b ¯ | 2 ) | a c ¯ + b a ¯ + c b ¯ | 2
1 9 1 36 | 2 + 3 i | 2 = 7 4 9
which is obviously not possible.
vulkanere64t6

vulkanere64t6

Beginner2022-05-17Added 2 answers

Hint to expert answer:
( | a | | c ¯ | + | b | | a ¯ | + | c | | b ¯ | ) 2 | a c ¯ + b a ¯ + c b ¯ | 2 , unless a , b , c are positive real

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