Problem 27 - Calculus for Biology and Medicine

Question

Let $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ (a) Show that $$ A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$ has the repeated eigenvalues \(\lambda_{1}=\lambda_{2}=1\). (b) Show that \(\left[\begin{array}{l}1 \\ 0\end{array}\right]\) and \(\left[\begin{array}{l}0 \\ 1\end{array}\right]\) are eigenvectors of \(A\) and that any vector \(\left[\begin{array}{l}c_{1} \\ c_{2}\end{array}\right]\) can be written as $$ \left[\begin{array}{l} c_{1} \\ c_{2} \end{array}\right]=c_{1}\left[\begin{array}{l} 1 \\ 0 \end{array}\right]+c_{2}\left[\begin{array}{l} 0 \\ 1 \end{array}\right] $$ (c) Show that $$ \mathbf{x}(t)=c_{1} e^{t}\left[\begin{array}{l} 1 \\ 0 \end{array}\right]+c_{2} e^{t}\left[\begin{array}{l} 0 \\ 1 \end{array}\right] $$ is a solution of \((11.35)\) that satisfies the initial condition \(x_{1}(0)=c_{1}\) and \(x_{2}(0)=c_{2} .\)

Step-by-Step Solution

Answer

The matrix A has eigenvalues \(\lambda_1 = \lambda_2 = 1\) with eigenvectors \([1, 0]^T\) and \([0, 1]^T\). The solution \(\mathbf{x}(t) = c_1 e^t [1, 0]^T + c_2 e^t [0, 1]^T\) satisfies the differential equation with initial conditions \(x_1(0) = c_1\), \(x_2(0) = c_2\).

Step 1: Find the Eigenvalues of the Matrix A

To find the eigenvalues of the matrix \(A\), we solve the characteristic equation: \(\det(A - \lambda I) = 0\). For \[A = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\], we have \[\det\left(\begin{bmatrix} 1 - \lambda & 0 \ 0 & 1 - \lambda \end{bmatrix}\right) = (1 - \lambda)^2 = 0.\] Thus, the eigenvalues are \(\lambda_1 = \lambda_2 = 1\).

Step 2: Find the Eigenvectors of the Matrix A

An eigenvector \(\mathbf{v}\) corresponding to the eigenvalue \(\lambda = 1\) satisfies \(A\mathbf{v} = \lambda \mathbf{v}\). For \( \lambda = 1 \), \(A - I = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\), implying any vector satisfies \((A - I)\mathbf{v} = 0\). Thus, eigenvectors \(\mathbf{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\) and \(\mathbf{v}_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix}\) are valid.

Step 3: Express Any Vector as a Linear Combination of Eigenvectors

Any vector \(\begin{bmatrix} c_1 \ c_2 \end{bmatrix}\) can be expressed as \(c_1 \begin{bmatrix} 1 \ 0 \end{bmatrix} + c_2 \begin{bmatrix} 0 \ 1 \end{bmatrix}\), since this is a simple way of representing vectors as linear combinations using the standard basis.

Step 4: Verify the Solution Satisfies the Differential Equation

The proposed solution is \(\mathbf{x}(t) = c_1 e^t \begin{bmatrix} 1 \ 0 \end{bmatrix} + c_2 e^t \begin{bmatrix} 0 \ 1 \end{bmatrix}\). Differentiate to find \(\frac{d\mathbf{x}}{dt} = c_1 e^t \begin{bmatrix} 1 \ 0 \end{bmatrix} + c_2 e^t \begin{bmatrix} 0 \ 1 \end{bmatrix}\). This matches the right-hand side of \[\frac{d\mathbf{x}}{dt} = A\mathbf{x}\] since \(A\mathbf{x} = \mathbf{x}\) for our solution.

Step 5: Check the Initial Condition

Set \(t=0\) in the solution \(\mathbf{x}(t) = c_1 e^t \begin{bmatrix} 1 \ 0 \end{bmatrix} + c_2 e^t \begin{bmatrix} 0 \ 1 \end{bmatrix}\). We get \(\mathbf{x}(0) = c_1 \begin{bmatrix} 1 \ 0 \end{bmatrix} + c_2 \begin{bmatrix} 0 \ 1 \end{bmatrix} = \begin{bmatrix} c_1 \ c_2 \end{bmatrix}\), satisfying the initial conditions \(x_1(0) = c_1\) and \(x_2(0) = c_2\).

Key Concepts

Eigenvectors

In linear algebra, eigenvectors are special vectors associated with a linear transformation described by a matrix. They are crucial for understanding how this transformation affects a multidimensional space.
When a matrix \( A \) operates on an eigenvector \( \mathbf{v} \), the direction of the eigenvector remains unchanged, only its magnitude may change.
An eigenvector equation is generally represented as \( A\mathbf{v} = \lambda\mathbf{v} \), where \( \lambda \) is the eigenvalue corresponding to the eigenvector \( \mathbf{v} \). This means applying the transformation \( A \) scales the eigenvector by a factor of \( \lambda \), but does not alter its direction.
Understanding eigenvectors is not only useful for theoretical mathematics but also applies to various fields like physics, computer science, and engineering. They are used to simplify complex systems or decompose matrices for more efficient computations.

Linear Algebra

Linear algebra is a foundational branch of mathematics that deals with vectors, matrices, and linear transformations. It provides a framework for solving systems of linear equations, which are prevalent in various scientific fields.
At the core of linear algebra are concepts such as:

• Vectors: Objects representing quantities with magnitude and direction.
• Matrices: Arrays of numbers representing linear transformations.
• Matrix Operations: Such as addition, multiplication, and finding inverses.
• Determinants: Scalar values that provide important properties of matrices, like singularity.

Linear algebra tools, like eigenvectors and eigenvalues, are indispensable in computational mathematics, physics, natural sciences, and much more.
For example, they are used in developing differential equations models, 3D graphics rendering, and optimizing engineering systems. Since linear transformations map geometric transformations, they are essential for structural analysis or producing realistic animations in computer graphics.

Differential Equations

Differential equations are mathematical equations that involve derivatives of a function. They express relationships between functions and their rates of change and are fundamental in describing dynamic systems.
There are different types of differential equations:

• Ordinary Differential Equations (ODEs): Involve functions of a single variable and their derivatives.
• Partial Differential Equations (PDEs): Involve functions of multiple variables and partial derivatives.

Solutions to differential equations provide critical insights in fields such as physics, engineering, biology, and economics.
They help model real-world phenomena like population dynamics, electrical circuits, mechanical vibrations, or heat distribution.
For example, in this exercise, the solution \( \mathbf{x}(t) = c_1 e^{t}\mathbf{v}_1 + c_2 e^{t}\mathbf{v}_2 \) represents how the system evolves over time. The terms \( e^t \) correspond to exponential growth controlled by the eigenvalues of the transformation matrix. Differential equations aid in predicting future states of systems based on their current behavior, making them indispensable in both theoretical and applied sciences.