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Let A E R"Xn. Show that < Ax,y >=< x, Ay >, Vx, y E R" iff A = A'. Show this inner product equality holds for A E Cxn iff A = A* %3D = (A)', the conjugate transpose of A.

Let A E R"Xn. Show that < Ax,y >=< x, Ay >, Vx, y E R" iff A = A'. Show this inner product equality holds for A E Cxn iff A = A* %3D = (A)', the conjugate transpose of A.

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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Let \( A \in \mathbb{R}^{n \times n} \). Show that \( \langle Ax, y \rangle = \langle x, Ay \rangle \), \(\forall x, y \in \mathbb{R}^n \) if and only if \( A = A^t \). Show this inner product equality holds for \( A \in \mathbb{C}^{n \times n} \) if and only if \( A = A^* = (\overline{A})^t \), the conjugate transpose of \( A \).
Transcribed Image Text:Let \( A \in \mathbb{R}^{n \times n} \). Show that \( \langle Ax, y \rangle = \langle x, Ay \rangle \), \(\forall x, y \in \mathbb{R}^n \) if and only if \( A = A^t \). Show this inner product equality holds for \( A \in \mathbb{C}^{n \times n} \) if and only if \( A = A^* = (\overline{A})^t \), the conjugate transpose of \( A \).
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