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This is false. In particular, $A^0$ need not be positive semidefinite. For example, take $n=3$, $K = \{1,2,3\}$, let $v(x)$ be the column vector with a $1$ at position $x$ and $-1$ elsewhere, and le …

There is such an $A$ if and only if $M\geq 5$. To see this, first note that the condition that $A$ be a convex combination of terms $yy^T$ each with trace $a$ is irrelevant. As long as $A$ is positi …

If I understand correctly, the anser is yes. A completely positive $n\times n$ matrix can always be viewed as the gram matrix of some vectors in the nonnegative orthant of some $R^k$ and vice versa. …

This problem and various related problems are known to be NP-hard to solve exactly, but there has been a lot of work on efficient approximations. See this wikipedia page or try googling things like " …

In some sense this is (part of) the theory of linear programming. If you want a reference for that, check out Bertsimas and Tsitsiklis' Introduction to Linear Optimization.

Perhaps this should be a comment but it is too long. The classic result used for existence is (Farkas' Lemma), though this gives a non-existence condition rather than an existence condition. It says …

For the purposes of this answer I will ignore the condition of constant column sums. You ask for a matrix $A$ with $A^TA = \Sigma$ and $A\geq 0$ element wise. Such a matrix need not exist. For exam …

Strictly speaking it doesn't make much sense to talk about defining a function on $S$ when you've explicitly said you don't know any of the elements of $S$. I'll assume rather that you have a sequenc …

If you are willing to replace the vectors $c$, $x$, and $y$ by the spaces $U_1 = \text{span}(c)$ and $U_2 = \text{span}(x,y)$, which in some sense doesn't change the problem, then one generalization i …

The key phrase to google is "sparsest vector".

Let $N=n$, $m=1$, $P_i = 1$ for all $i$, and let the $C_i$ be the standard unit vectors. Then the left hand side of $(\star\star)$ is the matrix whose diagonal entries are the inverses of the diagona …

If your constraints on the $d_i$ allow it and you are interested in numerical solutions, you can cast the problem as a semidefinite program. To do so, introduce scalar variables $t_1,\ldots, t_n$. L …

One reason linear algebra is so useful is that the basic notions, like rank, have so many equivalent definitions. Some are better for formulating problems, some for proving theorems, and some for doi …

Using the axiom of choice, one can show that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups. In particular, they are both vector spaces over $\mathbb{Q}$ and AC gives bases of thes …

The condition you want is exactly that the matrix multiplication map be locally open at the pair $(B,C)$. This is the topic of the recent paper Where is matrix multiplication locally open? by Behrend …