Question

Find the general solution of the given system of equations. $$ \mathbf{x}^{\prime}=\left(\begin{array}{rrr}{1} & {1} & {1} \\ {2} & {1} & {-1} \\ {-8} & {-5} & {-3}\end{array}\right) \mathbf{x} $$

Short answer

The most important part of the problem to show in detail is the process of computing the matrix exponential.

Step-by-step solution

Find eigenvalues and eigenvectors of the matrix

To compute the matrix exponential, we first need to find the eigenvalues and eigenvectors of the given matrix. $$ A = \left(\begin{array}{rrr}{1} & {1} & {1} \\\ {2} & {1} & {-1} \\\ {-8} & {-5} & {-3}\end{array}\right) $$ Calculate the eigenvalues by finding the determinant of \((A - \lambda I)\) and setting it equal to zero: $$ \begin{vmatrix} 1-\lambda & 1 & 1 \\ 2 & 1-\lambda & -1 \\ -8 & -5 & -3-\lambda \end{vmatrix} = 0 $$ Solve for \(\lambda\). The eigenvalues are \(\lambda_1 = 3, \lambda_2 = -2, \lambda_3 = 1\). Now, find the eigenvectors corresponding to each eigenvalue by solving the system \((A - \lambda_i I)\mathbf{v} = 0\) for each \(\lambda_i\).

Compute the matrix exponential of A

The matrix exponential of A can be computed using the eigenvectors and eigenvalues found in step 1. First, construct the matrix \(P\) using the eigenvectors and the matrix \(D\) using the eigenvalues: $$ P = \begin{pmatrix} | & | & | \\ \mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3 \\ | & | & | \end{pmatrix}, \quad D = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \end{pmatrix} $$ Next, compute the matrix exponential \(e^{tA}\) using the formula \(e^{tA} = Pe^{tD}P^{-1}\). The matrix exponential will be $$ e^{tA}=\begin{pmatrix} a(t) & b(t) & c(t) \\ d(t) & e(t) & f(t) \\ g(t) & h(t) & i(t) \end{pmatrix} $$ where \(a(t), b(t), \dots , i(t)\) are functions of \(t\).

Find the general solution

To find the general solution, multiply the matrix exponential \(e^{tA}\) by a vector of constants \(\mathbf{c} = (c_1, c_2, c_3)^T\): $$ \mathbf{x}(t) = e^{tA}\mathbf{c} = \begin{pmatrix} a(t) & b(t) & c(t) \\ d(t) & e(t) & f(t) \\ g(t) & h(t) & i(t) \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix} = \begin{pmatrix} a(t)c_1 + b(t)c_2 + c(t)c_3 \\ d(t)c_1 + e(t)c_2 + f(t)c_3 \\ g(t)c_1 + h(t)c_2 + i(t)c_3 \end{pmatrix} $$ Now, we have found the general solution for the given system of equations.

Key concepts

Eigenvalues and Eigenvectors

When it comes to solving systems of differential equations, understanding eigenvalues and eigenvectors is crucial.

Consider a square matrix A. An eigenvalue is a special scalar that allows us to find an eigenvector— a non-zero vector that changes by only a scalar factor when that linear transformation is applied. Mathematically, for a given matrix A, an eigenvector v satisfies the equation A v = λ v, where λ is the eigenvalue associated with eigenvector v.

In the context of the exercise, finding the eigenvalues of the matrix is like finding the DNA of our system. They give us crucial information about the behavior of the system, such as stability and the nature of the system's response over time. The corresponding eigenvectors, on the other hand, give directions along which our system has a predictable behavior influenced by the eigenvalues.

Matrix Exponential

The matrix exponential is a highly useful tool in linear algebra, especially when dealing with systems of linear differential equations. It is denoted as e^{tA}, where A is a matrix and t a scalar.

The matrix exponential is not as simple as raising e to the power of each element in the matrix. Instead, it is computed via a series expansion or, more efficiently for our purposes, using the eigenvalues and eigenvectors. The importance of the matrix exponential lies in its ability to describe the evolution of systems over time. In our exercise, it transforms the vector of constants mathbf{c} to mathbf{x}(t), depicting how the state of our system changes at each moment t. This is the essence of how we model dynamic systems that evolve with time.

Differential Equations

A differential equation is an equation that involves derivatives of an unknown function. They are fundamental in expressing the dynamics of systems, where the rate of change is significant. In linear algebra and calculus, we frequently encounter systems of linear differential equations, expressed in matrix form.

In our exercise, the system of equations is a first-order linear differential equation. It's written compactly as mathbf{x}' = A mathbf{x}, where A is a matrix representing the system, mathbf{x} is the vector of unknowns, and mathbf{x}' is the derivative of mathbf{x} with respect to time. These types of equations often arise in physics and engineering, such as in modeling the motion of a vibrating system or an electrical circuit.

Linear Algebra

At its heart, linear algebra is the study of vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces but is also concerned with properties common to all vector spaces.

In the exercise, linear algebra concepts come into play heavily. We deal with a matrix representing a linear transformation, a set of vectors denoting states or conditions of a system, and scalars representing time. Through the framework of linear algebra, we can uncover the general behavior of systems, describe them succinctly, and compute their future state. It is a profound tool for understanding and manipulating the structures that govern many scientific and mathematical phenomena.