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The sensitivity of a multiple eigenvalue of a matrix under perturbations can be measured by its Hölder condition number. Various extensions of this concept are considered. A meaningful notion of structured Hölder condition numbers is introduced, and it is shown that many existing results on structured condition numbers for simple eigenvalues carry over to multiple eigenvalues. The structures investigated in more detail include real, Toeplitz, Hankel, symmetric, skew-symmetric, Hamiltonian, and skew-Hamiltonian matrices. Furthermore, unstructured and structured Hölder condition numbers for multiple eigenvalues of matrix pencils are introduced. Particular attention is given to symmetric/skew-symmetric, Hermitian, and palindromic pencils. It is also shown how matrix polynomial eigenvalue problems can be covered within this framework.