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For x, y = C", x ‡ 0, explain why equality holds in the CBS inequality if and only if y = ax, where a = x*y/x*x. CBS= Cauchy- Schwartz Inequality

For x, y = C", x ‡ 0, explain why equality holds in the CBS inequality if and only if y = ax, where a = x*y/x*x. CBS= Cauchy- Schwartz Inequality

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 5.1.9:** For \( \mathbf{x}, \mathbf{y} \in \mathbb{C}^n \), \( \mathbf{x} \neq 0 \), explain why equality holds in the CBS inequality if and only if \( \mathbf{y} = \alpha \mathbf{x} \), where \( \alpha = \frac{\mathbf{x}^* \mathbf{y}}{\mathbf{x}^* \mathbf{x}} \). **Explanation:** CBS = Cauchy-Schwartz Inequality In this problem, we are examining when equality occurs in the Cauchy-Schwartz inequality for vectors in a complex vector space. The equality condition is satisfied when one vector is a scalar multiple of the other. Here, the scalar \( \alpha \) is defined as the quotient of the dot product \( \mathbf{x}^* \mathbf{y} \) over the self-dot product \( \mathbf{x}^* \mathbf{x} \). This highlights the concept that equality in the Cauchy-Schwartz inequality indicates parallel vectors in the context of complex vector spaces.
Transcribed Image Text:**Problem 5.1.9:** For \( \mathbf{x}, \mathbf{y} \in \mathbb{C}^n \), \( \mathbf{x} \neq 0 \), explain why equality holds in the CBS inequality if and only if \( \mathbf{y} = \alpha \mathbf{x} \), where \( \alpha = \frac{\mathbf{x}^* \mathbf{y}}{\mathbf{x}^* \mathbf{x}} \). **Explanation:** CBS = Cauchy-Schwartz Inequality In this problem, we are examining when equality occurs in the Cauchy-Schwartz inequality for vectors in a complex vector space. The equality condition is satisfied when one vector is a scalar multiple of the other. Here, the scalar \( \alpha \) is defined as the quotient of the dot product \( \mathbf{x}^* \mathbf{y} \) over the self-dot product \( \mathbf{x}^* \mathbf{x} \). This highlights the concept that equality in the Cauchy-Schwartz inequality indicates parallel vectors in the context of complex vector spaces.
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