Question is from subject Linear System Theory. Problem 5: Norm Preserving Flowsa) 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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Description: Question is from subject Linear System Theory. Problem 5: Norm Preserving Flowsa) 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

c) To prove:- If A is skew-symmetric matrix , Q=eAt is orthogonal. Proof:- Ais skew-symmetric matrix…

Answered: Prove that if A is skew-symmetric…

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

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.

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