A cone $K$ in $\Bbb R^n$ is said to be proper if $K$ is convex, closed, $K \cap (-K) = \{0\}$, and $K$ has non-empty interior. The dual of $K$ is $$ K^* = \{x \in \Bbb R^n : \langle x,y \rangle \geq 0 \text{ for all } y \in K\} . $$ The Lorentz cone is the set $$ \Cal L_+^n = \left\{x \in \Bbb R^n : x_n \geq 0 ,\, \sum_{i=1}^{n-1} x_i^2 \leq x_n^2 \right\} . $$ It is known that $\Cal L_+^n$ is self-dual proper ellipsoidal. Moreover, any proper cone $K$ in $\Bbb R^n$ is ellipsoidal if and only if $K = X(\Cal L_+^n)$ for some $n \times n$ invertible matrix $X$ [R.~J. Stern and H. Wolkowicz, SIAM J. Matrix Anal. Appl. {\bf 12} (1991), no.~1, 160--165; MR1082333]. In this paper, the properties of $\Cal L_+^n$-semipositive matrices are studied. An $n \times n$ matrix $A$ is said to be $\Cal L_+^n$-semipositive if there exists a vector $x \in \Cal L_+^n$ such that $Ax \in \roman{Int}(\Cal L_+^n)$, the interior of $\Cal L_+^n$. The authors prove that if $A$ is an $n \times n$ matrix whose $n$-th column belongs to $\Cal L_+^n$, and if the $k$-th column of $A$ belongs to $\roman{Int}(\Cal L_+^n)$ for some $k \ne n$, then $A$ is $\Cal L_+^n$-semipositive. In addition, the authors characterize the diagonal and orthogonal $\Cal L_+^n$-semipositive matrices. They prove that a diagonal matrix $D = \roman{diag}(d_1,\dots ,d_n)$ is $\Cal L_+^n$-semipositive if and only if $d_n > 0$. Also, an orthogonal matrix $A = (a_{ij})$ is $\Cal L_+^n$-semipositive if and only if $a_{nn} >0$. Finally, $\Cal L_+^n$-semipositive cones are studied. An ${n \times n}$ matrix $A$ is said to be $\Cal L_+^n$-monotone if $Ax \in \Cal L_+^n$ implies $x \in \Cal L_+^n$. Given an $n \times n$ matrix $A$, let ${K = \{x \in \Cal L_+^n : Ax \in \Cal L_+^n\}}$ and $S = \{x \in \Bbb R^n : Ax \in \Cal L_+^n\}$. The authors show that $A$ is $\Cal L_+^n$-monotone if and only if $A$ is invertible and $K=S$ is an ellipsoidal cone.