This question is about the use of eigenvalues in a specific application. The subject is Computer Vision and the topic is the Harris Corner detection method. The attached file is PDF document of slides that show the math in a bit more detail. In the slides, a corner is located by looking at the image brightness gradient in a region (say 5 x5 neighboring pixel) about each pixel in the image. The gradient value (formula) is D(u,v) = [u v]C[u v]T = constant. C is the co-variance matrix for the neighborhood about a given pixel. C is then diagonalized with eigenvalues, and they and their eigenvectors indicate the direction and strength of the brightness gradient. But I’m puzzled why eigenvalues are calculated at all. Isn’t calculating D(u,v) for each pixel enough? I am aware that eigenvalues provide a root to the homogeneous equation Ax=0, but why try and find roots at all in this situation? And what is the equation we trying to solve in this case?