Quadratic form in 3D
A linear application is defined by the image of the three basis vectors. Its matrix M in the canonical basis is composed of each of their coordinates, arranged in columns. The determinant of the matrix is the (algebraic) volume of the parallelepiped generated by these three sides.
Each point of the unit sphere is sent to an ellipsoid. Its axes are given by its singular value decomposition. When the matrix is symmetric, these are its eigenvectors.
Feel free to rotate the figure!
You can display the basis in order to modify the matrix, column by column, by moving the images of the basis vectors (Red, Green, Blue). You can also edit them by the Input field by entering the coordinates of M1, M2 and M3, the points defining the images of the basis vectors.
You can move point A and see its image MA.
You can also display the quadrics: the sphere and its image by the linear application. You also see the three unit vectors, sent on the three principal axes of the ellipsoid. Point B is easier to manipulate than point A because it is attached to the sphere. Observe when you align it with the orthogonal basis defined on the sphere how the image MB aligns with the principal axes.