How to show this matrix is diagonally dominant

Question

Let $d\in \Delta_n$ (the unit simplex: $\Delta_n=\{x\in R_+^n|\sum_{i=1}^nx_i=1$} ).

Show that the nxn matrix defined by $$A_{ij}= \begin{cases} d_i-d_i^2 , &\text{$i=j$} \\ -d_id_j, &\text{$i\neq j$} \end{cases}$$

Is positive semi-definite. The assignment recommends using the theorem that says "If A is diagonally dominant with non-negative diagonal entries, then A is positive semi-definite."

I can't figure out how to show that A is diagonally dominant, i.e:

$$\lvert{A_{ii}\rvert}\ge\sum_{j\ne i}\lvert A_{ij} \rvert$$

Can anyone help? Thank you so much.

Answer 1

0.) ignoring the hint with diagonal matrix $D = \text{diag}\big(\mathbf d\big)$
note: see 2.) at the end for the way with the hint

you have $A =D - D\mathbf {11}^TD = D^\frac{1}{2}\big(I-D^\frac{1}{2}\mathbf {11}^TD^\frac{1}{2}\big)D^\frac{1}{2}$

specializing to the nonsingular $D$ case, you have $\big(I-D^\frac{1}{2}\mathbf {11}^TD^\frac{1}{2}\big)$
is a matrix with all eigenvalues of 1 except a single eigenvalue of 0 -- why? So $A$ is congruent to this positive semidefinite matrix and the result follows.

for the case of singular $D$ consider the quadratic form
$\mathbf x^T A \mathbf x = \mathbf x^T D^\frac{1}{2}\big(I-D^\frac{1}{2}\mathbf {11}^TD^\frac{1}{2}\big)D^\frac{1}{2}\mathbf x = \mathbf y^T \big(I-D^\frac{1}{2}\mathbf {11}^T D^\frac{1}{2}\big)\mathbf y \geq 0$
with change of variables $\mathbf y:= D^\frac{1}{2}\mathbf x$
and we know
$\mathbf y^T \big(I-D^\frac{1}{2}\mathbf {11}^T D^\frac{1}{2}\big)\mathbf y \geq 0$ because $\big(I-D^\frac{1}{2}\mathbf {11}^T D^\frac{1}{2}\big) \succeq 0$

addendum
1.) the original post indicates that $d_i \geq 0$ and $\mathbf 1^T \mathbf d = 1$ but this seems to be the definition of the probability simplex not the unit simplex...
What is the difference between a unit simplex and a probability simplex?

2.) if we wanted to do this with (weak) diagonal dominance / Gerschgorin discs, we could observe that all diagonal components of $A$ are $\geq 0$ and all off diagonal components are $\leq 0$. Given this homogeneity it is enough to look at

$A \mathbf 1 = D\mathbf 1 - D\mathbf {11}^TD\mathbf 1 = D\mathbf 1 - D\mathbf 1 \big(\mathbf 1^TD\mathbf 1\big) = D\mathbf 1 - \beta D\mathbf 1 = (1- \beta)\cdot D\mathbf 1 \geq \mathbf 0$
for some $\beta \in [0,1]$ -- we are told that $\beta = 1$ so $A\mathbf 1 = \mathbf 0$ though as noted in (1) this doesn't seem to be the standard definition for unit simplex. In any case $A$ is real symmetric and the radius of each Gerschgorin disc $r_i \leq a_{i,i}\geq 0$ which proves all eigenvalues are real non-negative and hence $A \succeq 0$