Let A be an n ×n matrix and k a scalar. Consider the...

Let $A$ be an $n \times n$ matrix and $k$ a scalar. Consider the following two systems: \[ \begin{array}{l} \frac{d \vec{x}}{d t}=A \vec{x} \\ \frac{d \vec{c}}{d t}=k A \vec{c} \end{array} \] Show that if $\vec{x}(t)$ is a solution of system (I), then $\vec{c}(t)=\vec{x}(k t)$ is a solution of system (II). Compare the vector fields of the two systems.

Let $A$ be an $n \times n$ matrix and $k$ a scalar. Consider the following two systems:
\[
\begin{array}{l}
\frac{d \vec{x}}{d t}=A \vec{x} \\
\frac{d \vec{c}}{d t}=k A \vec{c}
\end{array}
\]
Show that if $\vec{x}(t)$ is a solution of system (I), then $\vec{c}(t)=\vec{x}(k t)$ is a solution of system (II). Compare the vector fields of the two systems.

Key Concepts

Linear Dynamical Systems

A linear dynamical system consists of a set of differential equations where the derivative of the state vector is given by a linear transformation of the state vector. These systems are characterized by constant coefficients when the matrix involved does not depend on time, leading to solutions that can often be expressed in terms of matrix exponentials. They serve as an important model in many areas of science and engineering due to their tractability and the wealth of mathematical tools available for studying their behavior.

Time Scaling in Differential Equations

Time scaling involves a modification of the time variable in a differential equation to understand how the dynamics change when time is reparameterized. This concept is central to the problem since replacing t with k*t effectively accelerates or decelerates the evolution of the system. Through this process, one can often relate solutions of one system to another by suitable transformations, preserving the essential structure of the dynamics while revealing how time dilation or contraction alters the rate of evolution.

Chain Rule in Differentiation

The chain rule is a fundamental concept in calculus that allows the differentiation of composite functions. In the context of this problem, it is used to relate the derivative of a function evaluated at a scaled time to the derivative of the original function. By applying the chain rule, one can show that differentiating x(k*t) with respect to time introduces a factor of k, which is crucial for proving that the transformed function satisfies the modified differential equation.

Vector Fields

A vector field assigns a vector to every point in a given space and is used to visually and mathematically represent the direction and magnitude of a system's instantaneous rate of change. Comparing the vector fields of the two systems highlights the effect of the time scaling on the dynamics. When time is rescaled, the essential structure of the vector field is preserved while the magnitude of the vectors is altered by a constant factor, reflecting the acceleration or deceleration of the system’s trajectories.

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