Prove that the general solution to a linear system 𝐮=Λ𝐮 with diagonal coefficient matrix Λ=diag(

step 1 The topic of this question is eigenvalues and eigenvectors. This question asks us to show that f

Prove that the general solution to a linear system 𝐮=Λ𝐮 with...

Prove that the general solution to a linear system $\mathbf{u}=\Lambda \mathbf{u}$ with diagonal coefficient matrix $\Lambda=\operatorname{diag}\left(\lambda_1, \ldots+\lambda_n\right)$ is given by $\mathbf{u}(t)=\left(c_1 e^{\lambda_1 t}, \ldots, c_n e^{\lambda_n t}\right)^T$.

Key Concepts

Exponential Matrix Solution

The matrix exponential, exp(A*t), is a key tool in solving linear systems of differential equations. When dealing with a diagonal matrix A, the exponential matrix is computed by exponentiating each diagonal entry individually. This operation yields a solution in the form u(t)=exp(A*t)c, which for diagonal A becomes a vector of exponentials, each scaled by an arbitrary constant, reflecting the natural evolution of each independent equation in the system.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are central to understanding linear transformations. For diagonal matrices, the eigenvalues are precisely the entries on the diagonal and the eigenvectors are the standard basis vectors. These properties directly influence the behavior of the system's solutions, as the rate of change in each independent direction is determined by the corresponding eigenvalue.

Diagonal Matrices

A diagonal matrix is one in which non-zero entries occur only on the main diagonal. In the context of differential equations, a diagonal coefficient matrix signifies that each equation in the system is independent of the others, allowing each coordinate function to be solved separately. This decoupling simplifies the system, as the solution for each variable is found by solving a scalar differential equation.

Linear Systems of Differential Equations

This concept pertains to systems where the evolution of a vector is governed by a linear transformation. A fundamental example is the system u' = A*u, where A is a constant matrix. Solving such systems usually involves finding functions that satisfy each component of the system, often simplifying the analysis when A has properties that make the system decoupled, such as being diagonal.

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