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Prove that if A is skew-symmetric matrir, Q = et is orthogonal.

Prove that if A is skew-symmetric matrir, Q = et is orthogonal.

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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Question is from subject Linear System Theory.

Problem 5: Norm Preserving Flows a) For a given time-variant system \(\dot{x}(t) = A(t)x(t)\), the adjoint system is defined as \(\dot{y}(t) = -A(t)^T y(t)\). Prove that the inner product of the states of these two systems remains the same, that is, there exists \(c \in \mathbb{R}\) such that \(\forall t \in \mathbb{R} : x(t)^T y(t) = c\). b) Assume \(A(t) = -A(t)^T, \forall t\). Prove that \(\exists c \in \mathbb{R}\) such that: \(\|x(t)\|_2 = c \forall t\). c) Prove that if \(A\) is a skew-symmetric matrix, \(Q = e^{At}\) is orthogonal.
Transcribed Image Text:Problem 5: Norm Preserving Flows a) For a given time-variant system \(\dot{x}(t) = A(t)x(t)\), the adjoint system is defined as \(\dot{y}(t) = -A(t)^T y(t)\). Prove that the inner product of the states of these two systems remains the same, that is, there exists \(c \in \mathbb{R}\) such that \(\forall t \in \mathbb{R} : x(t)^T y(t) = c\). b) Assume \(A(t) = -A(t)^T, \forall t\). Prove that \(\exists c \in \mathbb{R}\) such that: \(\|x(t)\|_2 = c \forall t\). c) Prove that if \(A\) is a skew-symmetric matrix, \(Q = e^{At}\) is orthogonal.
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