1. **State the problem:** We need to find the eigen-decomposition of the coefficient matrix $A$ of a system of linear equations and then use this decomposition to solve the system. 2. **Eigen-decomposition formula:** For a square matrix $A$, the eigen-decomposition is given by: $$A = PDP^{-1}$$ where $P$ is the matrix of eigenvectors and $D$ is the diagonal matrix of eigenvalues. 3. **Steps to find eigen-decomposition:** - Find eigenvalues $\lambda$ by solving $\det(A - \lambda I) = 0$. - For each eigenvalue, find the corresponding eigenvector by solving $(A - \lambda I)\mathbf{v} = 0$. - Form matrix $P$ with eigenvectors as columns and $D$ with eigenvalues on the diagonal. 4. **Using eigen-decomposition to solve $A\mathbf{x} = \mathbf{b}$:** - Write $A = PDP^{-1}$. - Then $PDP^{-1}\mathbf{x} = \mathbf{b}$. - Multiply both sides by $P^{-1}$: $DP^{-1}\mathbf{x} = P^{-1}\mathbf{b}$. - Let $\mathbf{y} = P^{-1}\mathbf{x}$ and $\mathbf{c} = P^{-1}\mathbf{b}$, so $D\mathbf{y} = \mathbf{c}$. - Since $D$ is diagonal, solve for $\mathbf{y}$ by dividing each component: $y_i = \frac{c_i}{d_{ii}}$. - Finally, find $\mathbf{x} = P\mathbf{y}$. 5. **Summary:** - Compute eigenvalues and eigenvectors. - Form $P$ and $D$. - Compute $\mathbf{c} = P^{-1}\mathbf{b}$. - Solve $D\mathbf{y} = \mathbf{c}$. - Compute $\mathbf{x} = P\mathbf{y}$. This method leverages the diagonalization of $A$ to simplify solving the system.