Spectral decomposition is a mathematical technique that expresses an operator or a matrix in terms of its eigenvalues and eigenvectors, allowing us to analyze its behavior in a more straightforward manner. This concept is crucial in quantum mechanics, as it enables the representation of quantum states and observables in Hilbert spaces, connecting to the foundational principles of linear algebra. Understanding spectral decomposition also allows for better insight into stationary states and energy eigenvalues, forming the backbone of quantum systems.
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