\end{aligned}$$ by lebesgue’s dominated convergence theorem. as a result, by using ( 145 ), we obtain that consequently, which implies the desired result. \(\square \) proof (proof of corollary 1) we follow the proof of theorem 1 presented in sect. 4 . the first step is...
Thus, the John ellipsoid theorem implies the existence of a “reducing matrix”, defined as follows. Definition 13 (Reducing matrix) If V is a p-nondegenerate matrix weight, then for every cube Q⊆Rn, there exists a positive definite, Hermitian d×d matrix RQp(V), called a reducing ...
Indeed, by using the spectral decomposition theorem on the initial correlation matrix, C0=ΓΛΓt (where Λ is the diagonal matrix of ordered eigenvalues of C0 and Γ is an orthogonal matrix whose columns are the associated eigenvectors), the principal components transform of X is Y=ΓtX, ...