Lagrange's Identity and Cauhchy-Schwarz Inequality for complex numbers

In summary, the conversation discusses the Cauchy-Schwarz Inequality and Lagrange's Identity in the context of complex analysis. It is mentioned that the Cauchy-Schwarz Inequality is significant in validating the geometric interpretation of complex numbers as vectors, while there is uncertainty about the importance and applications of Lagrange's Identity.
  • #1
rmcknigh
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I guess the best way to start this is by admitting that my conceptual understanding of the Cauchy-Schwarz Inequality and the Lagrange Identity, as the title suggests, is not as deep as it could be.

I'm working through Marsden's 3e "Basic Complex Analysis" and it contains a proof of the Cauchy Schwarz Inequality using what it calls a clever math trick- manipulating \(\displaystyle \sum_{k=1}^n\mid z_k-d\bar{w_k}\mid^2\) where \(\displaystyle z_k,w_k\in\mathbb{C} \ \forall \ k \ s.t. \ 1\leq k \leq n\) and \(\displaystyle d=\frac{\sum_{k=1}^n{z_kw_k}}{\sum_{k=1}^n{ \mid w_k\mid^2}}\). Then, as an exercise, we are asked to prove Lagrange's Identity and then deduce the Cauchy-Schwarz Inequality from it.

Although these aren't extremely difficult proofs, I don't understand what's so important about them that I can't find anything else in either the supplement or index of the Marsden text that uses these results at all .

I'm pretty sure that the Cauchy-Schwarz Inequality is important because it validates the definition of the angle between two complex vectors. Or perhaps restated, it solidifies the geometric interpretation of complex numbers as vectors.

But what about Lagrange's Identity? I'm having trouble finding applications of this to complex analysis besides a few different proofs of it, and its implication of the Cauchy Schwarz Inequality.

Halp.

Anthony
 
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  • #2
rmcknigh said:
I'm pretty sure that the Cauchy-Schwarz Inequality is important because it validates the definition of the angle between two complex vectors. Or perhaps restated, it solidifies the geometric interpretation of complex numbers as vectors.

But what about Lagrange's Identity? I'm having trouble finding applications of this to complex analysis besides a few different proofs of it, and its implication of the Cauchy Schwarz Inequality.
Hi Anthony and welcome to MHB!

The Cauchy–Schwarz inequality is absolutely central to the study of inner-product spaces. To take just one example, it gives the most natural proof of the triangle inequality $\|a+b\|\leqslant \|a\| + \|b\|$. But I have to admit that in 50 years of doing complex and functional analysis I have never until now come across Lagrange's inequality, so I'm not surprised that you are having trouble finding applications of it. According to Lagrange's identity - Wikipedia, the free encyclopedia, it crops up in exterior algebra, so it seems that maybe it is more of an algebraic than an analytical tool.
 

FAQ: Lagrange's Identity and Cauhchy-Schwarz Inequality for complex numbers

1. What is Lagrange's Identity for complex numbers?

Lagrange's Identity is a formula that relates the modulus (absolute value) of a complex number to the sum of the squares of its real and imaginary parts. It states that for any complex number, the square of its modulus is equal to the sum of the squares of its real and imaginary parts.

2. What is the Cauchy-Schwarz Inequality for complex numbers?

The Cauchy-Schwarz Inequality is a mathematical inequality that states that for any complex numbers a and b, the absolute value of their inner product (also known as the dot product) is less than or equal to the product of their respective moduli. In other words, the magnitude of the dot product of two complex numbers is always less than or equal to the product of their magnitudes.

3. What is the significance of Lagrange's Identity and Cauchy-Schwarz Inequality?

These identities are important tools in complex analysis and are frequently used in various fields of mathematics, physics, and engineering. They provide a way to relate the absolute value of a complex number to its real and imaginary parts, as well as provide a bound on the magnitude of the dot product of two complex numbers.

4. How are Lagrange's Identity and Cauchy-Schwarz Inequality related?

Lagrange's Identity can be seen as a special case of the Cauchy-Schwarz Inequality, where one of the complex numbers is taken to be the conjugate of the other. In other words, Lagrange's Identity can be derived from the Cauchy-Schwarz Inequality.

5. What are some applications of Lagrange's Identity and Cauchy-Schwarz Inequality?

These identities have various applications in mathematics, physics, and engineering. For example, they are used in optimization problems, signal processing, and in proving theorems in complex analysis. They also have applications in probability and statistics, where they are used to prove inequalities related to the covariance and correlation of random variables.

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