Eigenspaces are specific subspaces associated with a linear operator or matrix, defined for each eigenvalue as the set of all eigenvectors corresponding to that eigenvalue, along with the zero vector. These spaces play a crucial role in understanding the structure of operators, especially normal operators, and facilitate the decomposition of complex systems into simpler components. The spectral theorem highlights the significance of eigenspaces in representing operators in terms of their eigenvalues and corresponding eigenvectors.
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