Advanced Linear Algebra, Lecture 6.2: Spectral resolutions

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Advanced Linear Algebra, Lecture 6.2: Spectral resolutions

We begin this lecture by showing that every self-adjoint map H:X→X has only real eigenvalues, and an orthonormal basis of eigenvectors. As a result, it is diagonalizable by a unitary map U (i.e., a distance-preserving map, equivalently characterized by U*U=I). By the spectral theorem, X is a direct sum of H-eigenspaces, and the identity map can be written as a sum of the projection onto each eigenspace. This is called a resolution of the identity. Similarly, H can be written as a linear combination of these projection maps, called its spectral resolution. This allows us to easily define non-polynomial functions on H, as long as they are defined on the eigenvectors. This avoids the issue of convergence, which would arise if we tried to define something like the exponential function of H by plugging it into its Taylor series.