Experiments / live
Positive definite playground
Set the three entries of the symmetric matrix [[a, b], [b, c]] with the sliders below, and the contour map draws the quadratic form q(x) = a·x² + 2b·x·y + c·y² live. When both eigenvalues are positive the form is a bowl — nested closed contours around one lowest point; push the off-diagonal b until ac − b² crosses zero and the bowl tears open into a saddle. The readout shows the two eigenvalues in closed form and the badge gives the verdict. The full story — the five equivalent tests and why loss surfaces are bowls — is in the essay.
contour of q(x) = xᵀA x
Colour is height (ivory low → ink high); the heatmap floors negative values, so a saddle shows flat "dead" wedges where q < 0 beside the rising contoured wedges where q > 0.