Schwarz Inequality Proof

In summary, the conversation discusses an equation involving the quadratic term lambda and solving for its discriminant. It is noted that the Schwarz inequality should result in a "greater than or equal to" sign, but the manipulation leads to a "less than or equal to" sign. A solution is provided using a special value for lambda, but it is mentioned that this value may not be very interesting for all values of lambda.
  • #1
Old Guy
103
1

Homework Statement





Homework Equations


[itex](\langle \alpha | + \lambda^\ast\langle \beta |)(|\alpha\rangle+ \lambda|\beta\rangle) = \langle \alpha |\alpha \rangle + |\lambda|^2\langle \beta | \beta \rangle + \lambda \langle \alpha | \beta \rangle + \lambda^\ast \langle \beta | \alpha \rangle \geq 0[itex]



The Attempt at a Solution


This equation is the setup, and it leads to an equation that I can see is quadratic in lambda. From this, I calculate the discriminant, which must be greater than or equal to zero because all the terms are real and positive. However, when I manipulate this to get to the Schwarz inequality, I get a "less than or equal to" where I should have a "greater than or equal to". Can somone please help? Thanks.
 
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  • #2
Old Guy said:

Homework Statement


Homework Equations


[tex](\langle \alpha | + \lambda^\ast\langle \beta |)(|\alpha\rangle+ \lambda|\beta\rangle) = \langle \alpha |\alpha \rangle + |\lambda|^2\langle \beta | \beta \rangle + \lambda \langle \alpha | \beta \rangle + \lambda^\ast \langle \beta | \alpha \rangle \geq 0[/tex]

The Attempt at a Solution


This equation is the setup, and it leads to an equation that I can see is quadratic in lambda. From this, I calculate the discriminant, which must be greater than or equal to zero because all the terms are real and positive. However, when I manipulate this to get to the Schwarz inequality, I get a "less than or equal to" where I should have a "greater than or equal to". Can somone please help? Thanks.

Homework Statement


Homework Equations


The Attempt at a Solution


Use tex and /tex for the tex delimiters instead of latex and latex. And you don't want to solve a quadratic. Just put in the special value lambda=-<beta|alpha>/<beta|beta>.
 
Last edited:
  • #3
Thanks, and sorry about the equation format; I'm still trying to figure out the MathType translator.

Anyway, I understand how it works for the value of lambda you gave, but shouldn't it work for ANY value of lambda?
 
  • #4
Old Guy said:
Thanks, and sorry about the equation format; I'm still trying to figure out the MathType translator.

Anyway, I understand how it works for the value of lambda you gave, but shouldn't it work for ANY value of lambda?

It works for any lambda, but it doesn't tell you anything very interesting for every lambda. E.g. lambda=0 isn't interesting at all.
 

1. What is the Schwarz Inequality Proof?

The Schwarz Inequality Proof, also known as the Cauchy-Schwarz inequality, is a mathematical proof that states the relationship between the dot product of two vectors and their magnitudes. It shows that the absolute value of the dot product of two vectors is always less than or equal to the product of their magnitudes.

2. Why is the Schwarz Inequality Proof important?

The Schwarz Inequality Proof is important because it has various applications in mathematics, physics, and engineering. It is used to establish bounds and estimates in various mathematical problems, and it is also used in the proof of other theorems and inequalities.

3. How is the Schwarz Inequality Proof derived?

The Schwarz Inequality Proof is derived using the Cauchy-Schwarz inequality, which states that for any two vectors, the absolute value of their dot product is always less than or equal to the product of their magnitudes. This inequality is then used in the proof to show that the dot product of two vectors is maximized when they are parallel to each other.

4. What are some real-life examples of the Schwarz Inequality Proof?

The Schwarz Inequality Proof can be applied in various real-life situations, such as in physics to calculate the work done by a force on an object, in economics to determine the relationship between two variables, and in statistics to establish bounds on statistical measures.

5. Are there any variations of the Schwarz Inequality Proof?

Yes, there are various variations of the Schwarz Inequality Proof, such as the Cauchy-Schwarz-Bunyakovsky inequality, which extends the proof to include multiple vectors, and the Cauchy-Bunyakovsky-Schwarz inequality, which is a generalization of the Schwarz Inequality Proof to include integrals instead of just dot products.

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