Question

The matrix \(\Psi(t)\) is a fundamental matrix of the given homogeneous linear system. Find a constant matrix \(C\) such that \(\hat{\Psi}(t)=\Psi(t) C\) is a fundamental matrix satisfying \(\hat{\Psi}(0)=I\), where \(I\) is the \((2 \times 2)\) identity matrix. $$ \mathbf{y}^{\prime}=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right] \mathbf{y}, \quad \Psi(t)=\left[\begin{array}{cc} e^{t} & e^{t} \\ e^{-t} & -e^{-t} \end{array}\right] $$

Short answer

Question: Find the constant matrix \(C\) such that \(\hat{\Psi}(t)=\Psi(t) C\) is a fundamental matrix satisfying \(\hat{\Psi}(0)=I\), where \(\Psi(t)=\left[\begin{array}{cc} e^{t} & e^{t} \\\ e^{-t} & -e^{-t} \end{array}\right]\). Answer: The constant matrix \(C\) is given by \(C=\left[\begin{array}{cc} \frac{1}{2} & \frac{1}{2} \\\ \frac{1}{2} & \frac{1}{2} \end{array}\right]\).

Step-by-step solution

Write the given equations

We are given that $$ \Psi(t)=\left[\begin{array}{cc} e^{t} & e^{t} \\\ e^{-t} & -e^{-t} \end{array}\right] $$ and we need to find a constant matrix \(C\) such that \(\hat{\Psi}(t)=\Psi(t) C\) with \(\hat{\Psi}(0)=I\).

Define the matrices and equation to solve for C

Let \(C\) be a constant \(2 \times 2\) matrix: $$ C=\left[\begin{array}{cc} a & b \\\ c & d \end{array}\right] $$ Then, the product \(\Psi(t) C\) is given by: $$ \Psi(t) C = \left[\begin{array}{cc} e^{t} & e^{t} \\\ e^{-t} & -e^{-t} \end{array}\right]\left[\begin{array}{cc} a & b \\\ c & d \end{array}\right] $$

Find the product of the matrices

To find the product of the matrices, we use the matrix multiplication method: $$ \hat{\Psi}(t)= \left[\begin{array}{cc} e^{t}a+e^{t}c & e^{t}b+e^{t}d \\\ e^{-t}a-e^{-t}c & e^{-t}b-e^{-t}d \end{array}\right] $$

Apply the condition \(\hat{\Psi}(0)=I\) to find the constant matrix C

According to the given condition, the identity matrix is: $$ I = \left[\begin{array}{cc} 1 & 0 \\\ 0 & 1 \end{array}\right] $$ So, we set \(\hat{\Psi}(0)\) equal to the identity matrix and solve for each element in the constant matrix C: $$ \left[\begin{array}{cc} e^{0}a+e^{0}c & e^{0}b+e^{0}d \\\ e^{-0}a-e^{-0}c & e^{-0}b-e^{-0}d \end{array}\right] = \left[\begin{array}{cc} 1 & 0 \\\ 0 & 1 \end{array}\right] $$ Which gives us the following system of equations: $$ \begin{cases} a+c=1 \\ b+d=0 \\ a-c=0 \\ b-d=1 \end{cases} $$

Solve the system of equations

Solving the system of equations for a, b, c, and d, we get: $$ \begin{cases} a=c \\ b=1-d \\ a+c=1 \\ b-d=1 \end{cases} $$ By substituting the values, we find: $$ a=\frac{1}{2}, \quad b=d=\frac{1}{2}, \quad c=\frac{1}{2} $$

Write the resulting constant matrix

Now that we have the values for a, b, c, and d, we can write the constant matrix \(C\) as: $$ C=\left[\begin{array}{cc} \frac{1}{2} & \frac{1}{2} \\\ \frac{1}{2} & \frac{1}{2} \end{array}\right] $$ Thus, we have found the required constant matrix \(C\) such that \(\hat{\Psi}(t)=\Psi(t)C\) is a fundamental matrix satisfying \(\hat{\Psi}(0)=I\).

Key concepts

Homogeneous Linear Systems

Homogeneous linear systems are sets of linear equations in which all the constant terms are zero. As a result, these systems always have at least one solution: the trivial solution, where all variables are zero. For the equation \( \mathbf{y}^{\textprime} = A\mathbf{y} \), where \( A \) is a matrix of coefficients and \( \mathbf{y} \) is a vector of variables, we seek solutions that describe how \( \mathbf{y} \) behaves over time. When dealing with such systems, the fundamental matrix \( \Psi(t) \) comes into play. It's a solution to the matrix differential equation that helps us understand the behavior of the system over time. When constructing a fundamental matrix, we ensure it satisfies certain conditions, such as \( \hat{\Psi}(0) = I \), to help simplify initial value problems, where the system's status is known at \( t = 0 \).

Identity Matrix

The identity matrix, denoted as \( I \), is a special square matrix with ones on the main diagonal and zeros elsewhere. Its dimensions (\( n \times n \) for an \( n \)th order identity matrix) determine the size of the matrix. One of the key properties of the identity matrix is that when it is multiplied by any compatible matrix, the original matrix remains unchanged. This property comes into prominence when solving for the fundamental matrix in differential equations. Setting the value of \( \hat{\Psi}(t) \) equal to identity matrix at time \( t = 0 \) ensures that the resulting fundamental matrix satisfies initial value conditions—which is crucial in understanding the system's behavior at the initial state.

Matrix Multiplication

Matrix multiplication is a binary operation that takes a pair of matrices and produces another matrix. This operation is fundamental in linear algebra and appears frequently in solving systems of linear equations, transforming geometric figures, and many applications in engineering and science. Unlike scalar multiplication, matrix multiplication is not commutative—meaning the order of multiplication matters (\( AB \) does not necessarily equal \( BA \) ). To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The elements of the resulting matrix are calculated by summing the products of the corresponding entries from the rows of the first matrix and the columns of the second matrix. As seen in the step-by-step solution presented, matrix multiplication is a crucial step in finding the constant matrix \( C \) that transforms the given fundamental matrix \( \Psi(t) \) to satisfy initial conditions.

System of Equations

A system of equations is a set of two or more equations with a common set of variables. In linear algebra, these systems can be solved using various methods such as substitution, elimination, and matrix operations. The goal is to find the values of the variables that satisfy all equations simultaneously. In the context of differential equations, systems often arise when applying conditions to find specific constants, like matrix \( C \) in our example. By equating \( \hat{\Psi}(0) \) to the identity matrix, we create a system of equations that can be solved to determine the values of the elements in matrix \( C \) that will fulfill the required initial conditions. Solving these systems is fundamental in finding specific solutions to differential equations that model various physical, economical, or engineering processes.