Kunze Linear Algebra · Problem 4.3.3

This problem explores the spectral decomposition of a diagonalizable matrix using the minimal polynomial and Lagrange interpolation. Since $A$ is diagonal with distinct eigenvalues $\{1, 2, 3\}$, the polynomial $p$ annihilates it, and the Lagrange projectors $E_i$ isolate each eigenspace. Computing these projectors and verifying their orthogonality properties reveals how to reconstruct $A$ as a weighted sum of projections onto its eigenspaces—the core idea behind diagonalization.

This problem explores the spectral decomposition of a diagonalizable matrix using the minimal polynomial and Lagrange interpolation. Since $A$ is diagonal with distinct eigenvalues $\{1, 2, 3\}$, the polynomial $p$ annihilates it, and the Lagrange projectors $E_i$ isolate each eigenspace. Computing these projectors and verifying their orthogonality properties reveals how to reconstruct $A$ as a weighted sum of projections onto its eigenspaces—the core idea behind diagonalization.