# Spectral Projections

Chapter

## Abstract

This chapter considers \(
A \in \mathcal{M}_N \left( \mathbb{C} \right)
\) and uses the Dunford integral formula to construct the spectral projections of ℂ Precisely, this chapter is structured as follows. Section 3.1 gives a universal estimate for the norm of the inverse of a matrix in terms of its determinant and its norm. From this estimate it will become apparent that the eigenvalues of

^{N}onto*N*[(*A*−*λI*)^{ν(λ)}] for every*λ*∈*σ*(*A*). It also shows that, for each*λ*∈*σ*(*A*), the algebraic ascent*ν*(*λ*) equals the order of*λ*as a pole of the associated resolvent operator$$
\begin{array}{*{20}c}
{\mathcal{R}\left( {z;A} \right): = \left( {zI - A} \right)^{ - 1} ,} & {z \in \rho \left( A \right): = \mathbb{C}\backslash \sigma \left( A \right)} \\
\end{array} .
$$

(3.1)

*A*are poles of the resolvent operator (3.1). The necessary analysis to show this feature will be carried out in Section 3.3. Section 3.2 gives a result on Laurent series valid for vector-valued holomorphic functions. Finally, Section 3.4 constructs the spectral projections associated with the direct sum decomposition (1.6), whose validity was established by the Jordan Theorem 1.2.1.## Keywords

Laurent Series Spectral Projection Invertible Matrix Resolvent Operator Removable Singularity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Birkhäuser Verlag AG 2007