Linear Algebra with Applications, 10th Edition

6.3 Diagonalization

In this section, we consider the problem of factoring an $n\times n$ matrix A into a product of the form $XD{X}^{-1}$, where D is diagonal. We will give a necessary and sufficient condition for the existence of such a factorization and look at a number of examples. We begin by showing that eigenvectors belonging to distinct eigenvalues are linearly independent.

Theorem 6.3.1

If ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k$ are distinct eigenvalues of an $n\times n$ matrix A with corresponding eigenvectors ${\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2,\dots ,{\mathbf{\text{x}}}_k$, then ${\mathbf{\text{x}}}_1,\dots ,{\mathbf{\text{x}}}_k$ are linearly independent.

Proof

Let r be the dimension of the subspace of ${ℝ}^n$ spanned by ${\mathbf{\text{x}}}_1,\dots ,{\mathbf{\text{x}}}_k$ and suppose that $r<k$. We may assume (reordering the ${\mathbf{\text{x}}}_i$’s and ${\lambda }_i$’s if necessary) that ${\mathbf{\text{x}}}_1,\dots ,{\mathbf{\text{x}}}_r$ are linearly independent. Since ${\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2,\dots ,{\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_{r+1}$ are linearly dependent, ...