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

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