Some FAQs during office hours are collected here:

• Q1-3: Do you want us to test these results on computer generated cases or want the mathematical proof?
  I want mathematical proofs.

• Q2: Matrices $A$ and $B$ are not symmetric!
  Note only entries in the upper triangular part ($j \ge i$) are defined for $A$ and $B$. Lower triangular entries are determined by symmetry.

• Hints for Q2?
  Statisticians have a powerful way to show a matrix is p.s.d. That is to show that the matrix is the covariance matrix of a random vector. For matrix $A$, we consider the order statistics of $n$ iid uniform random variables on [0, 1]. For matrix $B$, we consider random variables $Y_i = X_1 + \cdots X_i$, where $X_1, \ldots, X_n$ are independent random variables with variances $\sigma_1^2, \ldots, \sigma_n^2$ respectively. These two examples are due to Ingram Olkin (1985) A probabilistic proof of a theorem of Schur.

• Q4, Q5: Do you want us to code everything from scratch like writing triple loops for matrix multiplications and so on?
  No, I just want you to make educated choice of numerical linear algebra functions in R. For both Q4 and Q5, I expect no more than 20 lines of R code.

• Q4: What is size of $k$ compared to $n$?
  Let’s assume $n \gg k$.

• Q4: I see no way to avoid a loop here :(
  Probably you are right, I don’t see obvious way to avoid loop in this case too.