Download the Wolfram demonstration project "Phase Portraits, Eigenvectors, and Eigenvalues". https://demonstrations.wolfram.com/PhasePortraitsEigenvectorsAndEigenvalues/Set [math: c1=c2=K=1]c_1=c_2=K=1. For which choice of [math: J] do you get the following phase diagram?

Question

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The question asks: For which choice of J do you obtain the given phase diagram in the Wolfram demonstration? The provided answer is J = [[1, 0], [0, -1]].

First, consider what J does in a linear system dx/dt = Jx. The eigenvalues of J determine the stability along the corresponding eigenvectors, shaping the phase portrait.

Option: J = [[1, 0], [0, -1]]  
- This is a diagonal matrix with eigenvalues λ1 = 1 and λ2 = -1. The eigenvectors are the standard basis vectors along the x-axis (for λ1 = 1) and y-axis (for λ2 = -1).
- Along the x-axis, trajectories grow exponentially away from the origin because the eigenvalue is positive (unstable direction).
- Along the y-axis, trajectories decay toward the origin because the eigenvalue is negative (stable direction).
- The resulting phase portrait features a saddle point at the origin with distinct invariant manifolds along the coordinate axes. Points off the axes tend to be attracted toward the stable manifold and repelled along the unstable one, creating a typical saddle pattern.

Why this choice matches the diagram (and why the others would not):
- If J had both eigenvalues positive or both negative (e.g., [[a,0],[0,d]] with a and d same sign), you would observe either a node (all trajectories diverging or converging) rather than a saddle, which contradicts the cross-like separatrix structure seen in the figure.
- If J had off-diagonal components (nonzero b or c), the eigenvectors would generally rotate away from the coordinate axes, altering the orientation of the invariant manifolds and the separatrices. The diagram appears to indicate axis-aligned invariant manifolds consistent with a diagonal J.

Therefore, the choice J = [[1, 0], [0, -1]] yields the saddle-type phase portrait described, with one stable and one unstable direction, which aligns with the expected features of the diagram in the prompt.

Download the Wolfram demonstration project "Phase Portraits, Eigenvectors, and Eigenvalues". https://demonstrations.wolfram.com/PhasePortraitsEigenvectorsAndEigenvalues/Set [math: c1=c2=K=1]c_1=c_2=K=1. For which choice of [math: J] do you get the following phase diagram?Short answer

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