Problem 5 - Complex Analysis

Question

Suppose \(n \in \mathbb{N}\) and \(z_{\nu}, w_{\nu} \in \mathbb{C}\) for \(1 \leq \nu \leq n .\) Prove $$ \left|\sum_{\nu=1}^{n} z_{\nu} w_{\nu}\right|^{2}=\sum_{\nu=1}^{n}\left|z_{\nu}\right|^{2} \cdot \sum_{\nu=1}^{n}\left|w_{\nu}\right|^{2}-\sum_{1 \leq \nu<\mu \leq n}\left|z_{\nu} \bar{w}_{\mu}-z_{\mu} \bar{w}_{\nu}\right|^{2} $$ (the LAGRANGE Identity) and conclude from this the CAUCHY-SCHWARZ inequality in \(\mathbb{C}^{n}:\) $$ \left|\sum_{\nu=1}^{n} z_{\nu} w_{\nu}\right|^{2} \leq \sum_{\nu=1}^{n}\left|z_{\nu}\right|^{2} \cdot \sum_{\nu=1}^{n}\left|w_{\nu}\right|^{2}. $$

Step-by-Step Solution

Answer

The identity is proven using sum expansions and manipulation, leading to the Cauchy-Schwarz inequality by omitting non-negative cross-term squares.

Step 1: Understanding the LHS

We begin by examining the left-hand side of the identity, \( \left| \sum_{u=1}^{n} z_{u} w_{u} \right|^{2} \). This is equivalent to multiplying the sum by its complex conjugate: \( \left( \sum_{u=1}^{n} z_{u} w_{u} \right) \left( \sum_{\mu=1}^{n} \bar{z}_{\mu} \bar{w}_{\mu} \right) \). Expanding this product reveals cross-terms involving products of different indices.

Step 2: Expanding the Product

When we expand \( \left( \sum_{u=1}^{n} z_{u} w_{u} \right) \left( \sum_{\mu=1}^{n} \bar{z}_{\mu} \bar{w}_{\mu} \right) \), we obtain \( \sum_{u=1}^{n} \sum_{\mu=1}^{n} z_{u} w_{u} \bar{z}_{\mu} \bar{w}_{\mu} \). This sum includes diagonal terms of the form \( |z_{u}|^2 |w_{u}|^2 \) and off-diagonal terms \( z_{u} w_{u} \bar{z}_{\mu} \bar{w}_{\mu} \) for \( u eq \mu \).

Step 3: Separating Diagonal and Off-Diagonal Terms

Separate the terms as follows: the diagonal terms sum to \( \sum_{u=1}^{n} |z_{u}|^2 |w_{u}|^2 \), while the off-diagonal terms are \( \sum_{1 \leq u < \mu \leq n} (z_{u} w_{u} \bar{z}_{\mu} \bar{w}_{\mu} + z_{\mu} w_{\mu} \bar{z}_{u} \bar{w}_{u}) \).

Step 4: Manipulating Off-Diagonal Terms

Consider each off-diagonal term: \((z_{u} \bar{w}_{\mu} - z_{\mu} \bar{w}_{u})(\bar{z}_{u} w_{\mu} - \bar{z}_{\mu} w_{u}) = |z_{u} \bar{w}_{\mu} - z_{\mu} \bar{w}_{u}|^2 \). Thus, the off-diagonal terms subtract their negative squares, simplifying to \( \sum_{1 \leq u < \mu \leq n} |z_{u} \bar{w}_{\mu} - z_{\mu} \bar{w}_{u}|^2 \).

Step 5: Lagrange Identity Conclusion

Using the above separations, the full expansion becomes: \( \sum_{u=1}^{n} |z_{u}|^2 |w_{u}|^2 - \sum_{1 \leq u < \mu \leq n} |z_{u} \bar{w}_{\mu} - z_{\mu} \bar{w}_{u}|^2 \). This is precisely the right-hand side of the Lagrange identity, thus proving the identity.

Step 6: Deriving Cauchy-Schwarz Inequality

To derive the Cauchy-Schwarz inequality, notice that \( |z_{u} \bar{w}_{\mu} - z_{\mu} \bar{w}_{u}|^2 \geq 0 \) for each \(u, \mu\). Therefore, \( \left| \sum_{u=1}^{n} z_{u} w_{u} \right|^{2} \leq \sum_{u=1}^{n}|z_{u}|^{2} \cdot \sum_{u=1}^{n}|w_{u}|^{2} \), the right-hand side of the Lagrange identity without negative terms.

Key Concepts

Complex Numbers

Complex numbers are numbers that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\). They extend the concept of one-dimensional real numbers to the two-dimensional complex plane.
Each complex number has a real part (\(a\)) and an imaginary part (\(b\)). Here are some key properties and operations involving complex numbers:

• Addition: Add the real parts together and the imaginary parts together: \((a + bi) + (c + di) = (a+c) + (b+d)i\).
• Multiplication: Apply the distributive property: \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\).
• Complex Conjugate: For a complex number \(z = a + bi\), its conjugate is \(\bar{z} = a - bi\). Multiplying a complex number by its conjugate results in a real number: \(z\bar{z} = a^2 + b^2\).
• Magnitude: The magnitude, or modulus, of \(z = a + bi\) is \(|z| = \sqrt{a^2 + b^2}\).

Understanding these basic operations and properties is essential in dealing with complex numbers as you explore deeper mathematical proofs and concepts, such as Lagrange identities and inequalities.

Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality is a fundamental concept in mathematics, especially in linear algebra, analysis, and related fields. It provides a bound on the dot product of two vectors in terms of their magnitudes.
In the context of complex numbers or vectors in \(\mathbb{C}^n\), the Cauchy-Schwarz inequality states:\[\left|\sum_{u=1}^{n} z_{u} w_{u}\right|^{2} \leq \sum_{u=1}^{n}\left|z_{u}\right|^{2} \cdot \sum_{u=1}^{n}\left|w_{u}\right|^{2}\]Here, \(z_{u}\) and \(w_{u}\) are complex numbers, and the sums represent complex inner products.
Some important insights about the Cauchy-Schwarz inequality include:

• Non-negativity: Both sides of the inequality are non-negative due to the properties of absolute values and squares.
• Equality Condition: The equality holds if and only if the vectors are linearly dependent, meaning one is a scalar multiple of the other.
• Application in Geometry: It is often used to determine the angle between two vectors. If the inequality reaches equality, the vectors have the same or opposite direction.

Understanding this inequality allows you to strategize and analyze different kinds of problems, from pure mathematics to applied physics, by setting necessary bounds for expressions involving inner products.

Mathematical Proofs

Mathematical proofs are logical arguments that demonstrate the truth of a given statement or theorem. Proofs are the heart of mathematics, ensuring that conclusions are reached through irrefutable logic.
Creating a mathematical proof involves several steps:

• Understanding the Statement: Clearly define what needs to be proven. In our original exercise, this involves demonstrating the Lagrange Identity and subsequently the Cauchy-Schwarz inequality.
• Breaking Down the Problem: Often, a proof requires decomposing the problem into smaller, manageable parts – as we do in Lagrange Identity by separating diagonal and off-diagonal terms.
• Logical Steps: Use a series of logical deductions, applying known results like properties of complex numbers, to arrive at the result. Induction, contradiction, contrapositive, and direct proof are common techniques.
• Conclusiveness: Ensure each step is justified and the conclusion logically follows the assumptions.

Proofs not only verify existing knowledge but also contribute to new discoveries by building upon established theorems and principles. Mastery of proof-writing enhances critical thinking and problem-solving skills needed across all areas of mathematics.