an improper node
An improper node is the in-between case of a node — the picture you get when a node is squeezed to the brink of becoming something else. Trajectories still flow straight in (or straight out) without circling, so it is genuinely a node, but they all crowd in along a single direction rather than spreading between two. Imagine combing hair: every strand sweeps into the part from the same side, all tangent to one line.
It arises in a linear system when the matrix has a REPEATED real eigenvalue (the two eigenvalues coincide) but only ONE independent eigenvector — the matrix is 'defective', it cannot be diagonalized. With just one eigenvector direction available, there is only one straight-line trajectory; every other trajectory bends around and comes into the equilibrium tangent to that same single line. The missing second direction is patched in the algebra by a generalized eigenvector, which produces the characteristic curved, all-one-tangent flow. If the repeated eigenvalue is negative the improper node is stable; if positive, unstable.
The improper node sits exactly on the boundary in the trace-determinant plane where a node is about to turn into a spiral (the curve where the discriminant is zero). It is the borderline case, so it is structurally delicate: the slightest change to the coefficients can tip it into a true two-tangent node on one side or a spiral on the other. Recognising it tells you the system is poised right at that transition.
The matrix [-1, 1; 0, -1] has the repeated eigenvalue -1 but only one eigenvector (along the x-axis). The origin is a stable improper node: every trajectory decays in, all coming in tangent to that single x-axis direction.
A repeated eigenvalue with only one eigenvector gives the single-tangent improper node.
A repeated eigenvalue alone does not make a node improper — it depends on the eigenvectors. With one independent eigenvector you get this improper node; with two (a multiple of the identity) you get the star node instead. Same eigenvalue, two very different pictures.