Geometry and algebra of linear maps, vectors, and matrices
A quadratic form xᵀAx takes a vector and returns a single number that varies quadratically in x, the matrix version of ax². As x ranges over all directions, this number traces out a surface, and the matrix A (taken symmetric) decides the shape of that surface.
Its eigenvalues are the curvatures along the eigen-axes, and their signs classify the surface completely. All positive: a bowl opening up (positive definite) with a clean minimum. All negative: a dome (negative definite) with a maximum. Mixed signs: a saddle (indefinite), up in some directions, down in others.
Toggle the eigenvalue signs in the figure and watch the surface morph between bowl, dome, and saddle.