  Given a proper cone $K$ in the Euclidean space $\mathbb{R}^n$, a square
matrix $A$ is said to be $K$-semipositive if there exists an $x\in K$ such that
$Ax\in \text{int}(K)$, the topological interior of $K$. The paper aims to study
algebraic and geometrical properties of $K$-semipositive matrices with special
emphasis on the self-dual proper Lorentz cone $\mathcal{L}^n_+=\{x\in
\mathbb{R}^n:x_n\geq 0,\sum\limits_{i=1}^{n-1}x_{i}^2\leq x_n^2\}$. More
specifically, we discuss a few necessary and other sufficient algebraic
conditions for $\mathcal{L}^n_+$-semipositive matrices. Also, we provide
algebraic characterizations for diagonal and orthogonal
$\mathcal{L}^n_+$-semipositive matrices. Furthermore, given a square matrix $A$
and a proper cone $K$, geometric properties of the semipositive cone
$\mathcal{K}_{A,K}=\{x\in K:~Ax\in K\}$ and the cone of
$\mathcal{S}_{A,K}=\{x:Ax\in K\}$ are discussed in terms of their extremals. As
$\mathcal{L}^n_+$ is an ellipsoidal cone, at last we find results for the cones
$\mathcal{K}_{A,\mathcal{L}^n_+}$ and $\mathcal{S}_{A,\mathcal{L}^n_+}$ to be
ellipsoidal.
