Hadamard products and multivariate statistical analysis

Nonlinear Principal Component Analysis

“…The eigenvalues of the MCA problem are those of E 1 and those of E 2 and so on. The largest nontrivial one is the largest eigenvalue of E 2 , the smallest nontrivial one is the smallest of E 2 [Styan, 1973]. but the order of the others is undecided.…”

Schur multiplier characterization of a class of infinite matrices

“…There is, however, much justification for the term "Schur product" and we refer the reader to [2] and [20] for an historical discussion. This concept was first investigated by Schur in his paper [17] and has since appeared in several different areas of analysis: [15], [18], [19](complex function theory); [1], [12] (Banach spaces); [21], [14], [4] (operator theory); [5], [3] (matriceal harmonic analysis) and [20] (multivariate analysis).…”

“…This concept was first investigated by Schur in his paper [17] and has since appeared in several different areas of analysis: [15], [18], [19](complex function theory); [1], [12] (Banach spaces); [21], [14], [4] (operator theory); [5], [3] (matriceal harmonic analysis) and [20] (multivariate analysis). If X and Y are two Banach spaces of matrices we define Schur multipliers from X to Y as the space M (X, Y ) = {M : M * A ∈ Y for every A ∈ X}, equipped with the natural norm M = sup…”

Flexible Multivariate GARCH Modeling with an Application to International Stock Markets

“…The Hadamard product of two positive semi-definite matrices is positive semi-definite; for example, see Styan (1973). In addition, the sum of two positive semi-definite matrices is positive semi-definite.…”