On the conditional distribution of a multivariate Normal given a transformation – the linear case

Abstract

We show that the orthogonal projection operator onto the range of the adjoint T⁎ of a linear operator T can be represented as UT, where U is an invertible linear operator. Given a Normal random vector Y and a linear operator T, we use this representation to obtain a linear operator trueTˆ such that trueTˆY is independent of TY and Y−trueTˆY is an affine function of TY. We then use this decomposition to prove that the conditional distribution of a Normal random vector Y given scriptTY, where scriptT is a linear transformation, is again a multivariate Normal distribution. This result is equivalent to the well-known result that given a k-dimensional component of a n-dimensional Normal random vector, where k<n, the conditional distribution of the remaining true(n−ktrue)-dimensional component is a true(n−ktrue)-dimensional multivariate Normal distribution, and sets the stage for approximating the conditional distribution of Y given g(Y), where g is a continuously differentiable vector field.

Thiosemicarbazone ligands based transition metal complexes: A multifaceted investigation of antituberculosis, anti‐inflammatory, antibacterial, antifungal activities, and molecular docking, density functional theory, molecular electrostatic potential, absorption, distribution, metabolism, excretion, and toxicity studies

“…Conversely, “Softness,” inversely related to hardness, illustrates reactivity, with (9) exhibiting the highest softness, indicating increased reactivity. Compound (7) is notably the most electrophilic, indicating a strong tendency to accept electrons 15,46 …”

“…In Figure 7, the blue color‐coded region signifies the most positive electrostatic potential, pinpointing an electron‐deficient zone 44 . Conversely, the red color‐coded region represents the most electronegative potential, highlighting an electron‐rich domain 14,46 …”

“…Compound ( 7) is notably the most electrophilic, indicating a strong tendency to accept electrons. 15,46 These descriptors offer insights into chemical behavior, electronic properties, and potential applications in various fields. Higher hardness implies stability, while greater softness indicates higher reactivity, pivotal in predicting chemical reactions.…”

“…44 Conversely, the red color-coded region represents the most electronegative potential, highlighting an electron-rich domain. 14,46 For the ligand (2), a detailed examination of the MESP reveals that the most positive electrostatic potential localizes to specific atoms, including C7, H1, H2, H3, H4, H6, H7, H10, H11, H12, H13, H15, and H17. In striking contrast, the most electronegative potential congregates around atoms like Br1, O2, and S1.…”

Hydrogels Loaded with Atorvastatin‐Metal Organic Framework Have a Preventive Effect on Coronary Heart Disease

“…Lately, transitional metallic complexes that contain inorganic metallic ions and organic ligands are in focus due to their structure diversity, but more so due to their multiple applications for instance anti‐tumor and anti‐bacterial activities, insulin mimetic activity, luminescent characteristics, catalytic activity, magnetism, DNA conjugation characteristics, adsorption and removal of harmful substances, as well as gas adsorption capabilities [4–15] . As is commonly known, the structural diversity and active characteristics of metal‐organic complexes greatly rely on the selection of metal atoms and organic ligands employed, and reaction mechanism used to obtain the complexes.…”