Spectral Theorem for Diagonalizable Matrices

It occurs to me that most presentation of the spectrum theorem only concerns orthonormal basis.  This is a more general result from Meyer.

Theorem

A matrix $\mathbf{A} \in \mathbb{R}^{n\times x}$ with spectrum $\sigma(\mathbf{A}) = \{ \lambda_1, \dotsc, \lambda_k \}$ is diagonalizable if and only if there exist matrices $\{ \mathbf{G}_1, \dotsc, \mathbf{G}_k\}$ such that  $$\mathbf{A} = \lambda_1 \mathbf{G}_1 + \dotsb + \lambda_k  \mathbf{G}_k$$ where the $\mathbf{G}_i$'s have the following properties

• $\mathbf{G}_i$ is the projector onto $\mathcal{N} (\mathbf{A} - \lambda_i \mathbf{I})$ along $\mathcal{R} ( \mathbf{A} - \lambda_i \mathbf{I} )$. 
• $\mathbf{G}_i\mathbf{G}_j = 0$ whenever $i \neq j$
• $\mathbf{G}_1 + \dotsb + \mathbf{G}_k = 1$

The expansion is known as the spectral decomposition of $\mathbf{A}$, and the $\mathbf{G}_i$'s are called the spectral projectors associated with $\mathbf{A}$.

Note that being a projector $\mathbf{G}_i$ is idempotent.

• $\mathbf{G}_i = \mathbf{G}_i^2$

And since $\mathcal{N}(\mathbf{G}_i) = \mathcal{R}(\mathbf{A} - \lambda_i \mathbf{I} )$ and $\mathcal{R}(\mathbf{G}_i) = \mathcal{N}(\mathbf{A} - \lambda_i \mathbf{I} )$, we have the following equivalent complimentary subspaces

• $\mathcal{R}(\mathbf{A} - \lambda_i \mathbf{I} ) \oplus \mathcal{N}(\mathbf{A} - \lambda_i \mathbf{I} )$
• $\mathcal{R}(\mathbf{G}_i) \oplus \mathcal{N}(\mathbf{A} - \lambda_i \mathbf{I} )$
• $\mathcal{R}(\mathbf{A} - \lambda_i \mathbf{I} ) \oplus  \mathcal{N}(\mathbf{G}_i)$
• $\mathcal{R}(\mathbf{G}_i)  \oplus  \mathcal{N}(\mathbf{G}_i)$