The Hadamard product \([a_{ij}]\circ\) \([b_{ij}]\)of two \(m\times n\) matrices is the entrywise product \([a_{ij}b_{ij}].\) Let \(A_{1},\dots,A_{k}\) be positive semidefinite \(n\times n\) matrices and let \(R\) be the range of \(A_{1} \circ\dots\circ A_{k}\). The authors prove that \(R\) is spanned by the set \(\{A_{1}x_{1}\circ\dots\circ A_{k}x_{k}~| ~x_{1},\dots,x_{k}\in\mathbb{C}^{n} \}\), and even by the smaller set \(\{A_{1}x\circ\dots\circ A_{k} x~| ~x\in\mathbb{C}^{n}\}\). As a corollary, for any complex \(n\times n\) matrices \(B_{1},\dots,B_{k}\), the range of \(B_{1}B_{1}^{\ast}\circ\dots\circ B_{k}B_{k}^{\ast}\) is spanned by \(\{B_{1}x_{1}\circ\dots\circ B_{k}x_{k}~| ~x_{1},\dots,x_{k}\in\mathbb{C}^{n}\}\) (but not by the smaller set with the \(x_{i}\) all equal). This generalizes a result of \textit{X. Sun, X. Du} and \textit{D. Liu} [Linear Algebra Appl. 416, No. 2--3, 868--871 (2006; Zbl 1106.15020)].