Hopf bifurcation for dynamical systems $\dot x = f(x,\lambda )$ is characterized by rank deficiency two of $A^2 + \omega ^2 I$, where $A = f_x (x,\lambda ) \in \mathbb{R}^{n \times n} $ is the Jacobian at the Hopf point and $ \pm i\omega $ are the Hopf eigenvalues. Replacing $w^2 $ by $\nu $ and allowing $\nu \leq 0$ we introduce real and imaginary Hopf matrices A with Hopf numbers $\nu $, defining a codimension-2 manifold $\mathcal {M}_{H,\nu } \subset \mathbb{R}^{n \times n} \times \mathbb{R}$, which can be described by two scalar equations $\alpha (A,\nu ) = 0,\beta (A,\nu ) = 0$ given by the solution of certain linear bordered systems. These two scalar equations form a suitable augmentation of $f(x,\lambda ) = 0$ for the numerical computation of branches of Hopf points in two-parameter systems. In contrast to [Chu, Govaerts, and Spence, SIAM J. Numer. Anal., 31 (1994), pp. 524–539] we suggest an A-dependent bordering having certain advantages. We construct a scalar test function for Hopf points (in the sense of [Werner, in Bifurcation and Symmetry, Internat. Ser. Numer. Math., 104 (1992), pp. 317–327]) by nonlinear elimination of v from the two scalar equations $\alpha = 0$ and $\beta = 0$. A strict sign change of this test function during numerical path following of equilibria is shown to be equivalent to eigenvalue crossing conditions in (real or imaginary [Werner and Janovsky, in Bifurcation and Chaos, Internat. Ser. Numer. Math., 97 (1991), pp. 377–388]) Hopf bifurcation points and can hence be used for the detection of Hopf points. An extension to Hopf points for mappings (Naimark–Sacker bifurcation) is given. Finally we illustrate our method by several examples, particularly for the computation of Hopf curves in two-parameter families of vector fields.