Let \(K\) be a proper cone in \(\mathbb{R}^n\), and \(A\) a given \(n\times n\) real matrix. \(A\) is called \(K\)-nonnegative iff \(AK\subset K\), \(K\)-positive, if \(A(K\setminus \{0\})\subset \overset \circ K\). \(K\)-irreducible, if \(AK\subset K\) and no eigenvector of \(A\) is in \(\partial K\), and strongly \(K\)-nonnegative if \(AK\subseteq K\) and there exists an eigenvector in \(\overset \circ K\). The authors show here that whenever there is a proper cone \(K\) such that \(A\) is \(K\)-positive, \(K\)-irreducible or strongly \(K\)-nonnegative, then there is also such an ellipsoidal cone. This is done using well-known equivalent conditions on the spectrum of \(A\). It is shown that this does not hold for the property of \(K\)-nonnegativity. A spectral condition which is shown to be equivalent to the existence of an ellipsoidal cone \(K\) such that \(A\) is \(K\)-nonnegative, is given and it is stronger than the condition for general proper cones.