Question

Find the general solution. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rrr} 5 & -1 & 1 \\ -1 & 9 & -3 \\ -2 & 2 & 4 \end{array}\right] \mathbf{y} $$

Short answer

Given the matrix A: $$ A = \begin{bmatrix} 5 & -1 & 1 \\ -1 & 9 & -3 \\ -2 & 2 & 4 \end{bmatrix} $$ Find the general solution for the following system of linear equations: $$ \begin{cases} y_1^\prime = 5y_1 - y_2 + y_3 \\ y_2^\prime = -y_1 + 9y_2 - 3y_3 \\ y_3^\prime = -2y_1 + 2y_2 + 4y_3 \end{cases} $$ Follow the steps mentioned in the solution to find the eigenvalues, eigenvectors, construct a fundamental matrix, and finally, find the general solution.

Step-by-step solution

Determine the eigenvalues

To find the eigenvalues (\(\lambda\)) of the matrix \(A\), we need to compute the characteristic equation \(|A - \lambda I| = 0\), where \(I\) is the identity matrix. For the given matrix \(A\), the characteristic equation is: $$ \begin{vmatrix} 5-\lambda & -1 & 1 \\ -1 & 9-\lambda & -3 \\ -2 & 2 & 4-\lambda \end{vmatrix} = 0 $$ Now, calculate the determinant to find the eigenvalue equation: $$ (5-\lambda)[(9-\lambda)(4-\lambda) - (-3)(2)] - (-1)[(-1)(4-\lambda) - (-3)(2)] + 1[(-1)(2) - (-2)(9-\lambda)] = 0 $$ Solve this polynomial equation to find the eigenvalues \(\lambda\).

Find the eigenvectors

Next, we need to find the eigenvectors corresponding to each eigenvalue. For each eigenvalue \(\lambda\), we can find the eigenvectors by solving the following system of linear equations: \((A - \lambda I)\mathbf{v} = 0\), where \(\mathbf{v}\) is an eigenvector. Solve this system for each eigenvalue to find the associated eigenvectors.

Construct a fundamental matrix

Once we have all the eigenvalues and eigenvectors, we can construct a fundamental matrix \(X(t)\) that is formed by the eigenvectors. The columns of the fundamental matrix are given by the eigenvectors multiplied by the exponential of the eigenvalue times the independent variable (\(t\)): $$ X(t) = \begin{bmatrix} v_{1, 1}e^{\lambda_1 t} & v_{1, 2}e^{\lambda_2 t} & v_{1, 3}e^{\lambda_3 t} \\ v_{2, 1}e^{\lambda_1 t} & v_{2, 2}e^{\lambda_2 t} & v_{2, 3}e^{\lambda_3 t} \\ v_{3, 1}e^{\lambda_1 t} & v_{3, 2}e^{\lambda_2 t} & v_{3, 3}e^{\lambda_3 t} \end{bmatrix} $$ where \(v_{i, j}\) is the \(i\)-th component of the \(j\)-th eigenvector.

Find the general solution

Finally, we can find the general solution \(\mathbf{y}(t)\) by multiplying the fundamental matrix \(X(t)\) by a vector of constants \(\mathbf{c}\): $$ \mathbf{y}(t) = X(t) \mathbf{c} $$ The general solution will be a linear combination of the columns of the fundamental matrix, with the coefficients given by the vector \(\mathbf{c}\).

Key concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is essential in solving differential equations involving matrices, which crop up in various branches of mathematics and physics. Eigenvalues, denoted by \(\lambda\), are scalars associated with a matrix. They are found by solving the characteristic equation \(\det(A - \lambda I) = 0\), where \(A\) is the given matrix and \(I\) is the identity matrix.

Once the eigenvalues are found, eigenvectors can be determined. These are non-zero vectors \(\mathbf{v}\) that satisfy the equation \( (A - \lambda I)\mathbf{v} = 0\). In the context of differential equations, each eigenvector corresponds to a solution that grows or decays at a rate proportional to its eigenvalue. Therefore, calculating each eigenvalue and its respective eigenvector allows us to build the solutions to the system of equations.

To facilitate understanding, visualize a stretching or compressing transformation in geometry. The directions in which these stretches or compresses occur are the eigenvectors, and the factors by which they are stretched or compressed are the eigenvalues.

Fundamental Matrix

The fundamental matrix is a cornerstone concept for solving systems of linear differential equations. Once you have eigenvalues and the corresponding eigenvectors, the next step is to form a fundamental matrix \(X(t)\). This matrix serves as a template to find the general solutions, and it’s constructed using eigenvectors and their related eigenvalues.

The columns of the fundamental matrix are built from the eigenvectors, with each column vector multiplied by the exponential function of the corresponding eigenvalue times the independent variable \(t\). This form \(X(t)\) reflects how the system evolves over time. In essence, the fundamental matrix represents a transition from one state to another and is crucial for understanding the dynamics of the system described by the differential equations.

Remember, the key to constructing a fundamental matrix lies in the interplay between algebra and calculus. Algebra provides us with the eigenvectors, and calculus comes into play when we introduce the exponential function, signifying change through time.

System of Linear Equations

Solving a system of linear equations is a fundamental skill in algebra that applies directly to finding eigenvectors in our differential equations problem. When we have an eigenvalue, we plug it into the equation \( (A - \lambda I)\mathbf{v} = 0\) and solve for the eigenvector \(\mathbf{v}\). This process involves a system of linear equations, which can have one solution, infinitely many solutions, or no solution.

For this exercise, it's crucial to find non-trivial solutions for eigenvectors—meaning the solutions that are not just the zero vector. These solutions form the basis for constructing our fundamental matrix. Techniques such as Gaussian elimination or matrix row operations are often employed to solve these systems.

Breaking down complex problems into systems of linear equations not only makes them more manageable but also leverages the powerful methods developed for linear algebra to analyze and solve a wide range of problems, from simple to highly multidimensional ones.