Quadratic inequalities for complex variables?

[mathsciguy]

Hello, I was looking at Riley's solution manual for this specific problem. Along the way, he ended up with a quadratic inequality:

View attachment how.bmp

If you looked at the image, he said it is given that λ is real, but he asserted that λ has no real roots because of the inequality. Doesn't that mean λ is imaginary or complex in some points then, contradicting his first statement? I reckon this has something to do with the properties of a quadratic inequality for complex variables.

 

[pasmith]

Hello, I was looking at Riley's solution manual for this specific problem. Along the way, he ended up with a quadratic inequality:

View attachment 61729

If you looked at the image, he said it is given that λ is real, but he asserted that λ has no real roots because of the inequality. Doesn't that mean λ is imaginary or complex in some points then, contradicting his first statement? I reckon this has something to do with the properties of a quadratic inequality for complex variables.

$\lambda$ doesn't have roots; it's an arbitrary real number. The quadratic $P: z \mapsto az^2 + bz + c$ has roots, which are those $z \in \mathbb{C}$ for which $P(z) = 0$. The point is that if $P(\lambda) > 0$ for all real $\lambda$ then $P$ has no real roots, because if $z$ is real then $P(z) \neq 0$ and $z$ cannot be a root of $P$.

 

[mathsciguy]

That's cool, I get it now. Then that means the roots are either purely imaginary or complex (but not purely real) right? Then why is it required that the discriminant be less than zero? Is it because it will make sure that part of the solution will have an imaginary part?

 

[pasmith]

That's cool, I get it now. Then that means the roots are either purely imaginary or complex (but not purely real) right? Then why is it required that the discriminant be less than zero? Is it because it will make sure that part of the solution will have an imaginary part?

Yes. A quadratic with real coefficients has no real roots if and only if the discriminant is negative.

 

[CellCoree]

Hello there,

Thank you for bringing this up. You are correct in your observation that the statement about λ being real and having no real roots may seem contradictory. However, in this context, we are dealing with complex variables and their properties, which can be different from real numbers.

In the case of quadratic inequalities for complex variables, the roots are not necessarily real numbers. They can be complex numbers, which can be expressed as a combination of a real and imaginary part. So, when we say that λ is real, it means that the real part of λ is a real number, but the imaginary part could still be non-zero.

In the context of this problem, the inequality is used to show that λ cannot have real roots, which means that the quadratic equation cannot be solved using real numbers. This does not necessarily mean that λ is purely imaginary, but rather that it has a non-zero imaginary part.

I hope this clarifies the concept of quadratic inequalities for complex variables. It is important to remember that the properties and solutions of complex numbers can be different from real numbers, and this is something that we need to keep in mind while working with them.

Best,