Oblique Projections

A projection matrix P satisfies P² = P (idempotence). Geometrically, this means applying the transformation a second time does not change the result, because the output is already in the target subspace. However, a projection does not have to be an orthogonal projection (where the matrix is symmetric and the residual x - Px is perpendicular to the range of P).

Explore oblique vs orthogonal projections

Open Oblique Projections. Choose the target line and direction to see how the residual and the projection matrix behave.

1. Set Range u1 = 1, Range u2 = 0, Direction v1 = 1, Direction v2 = 1. This projects onto the x-axis along the line y = x.
2. Set Vector x1 = 2, Vector x2 = 3. The projected vector is (-1, 0). The residual is (3, 3). Notice the residual is parallel to v, but not perpendicular to u. The "Orthogonal projection" status is false.
3. Change Direction v1 = 1, Direction v2 = -1, and set Range u1 = 1, Range u2 = 1. Observe that u and v are perpendicular.
4. With Vector x1 = 2, x2 = 0, the product Px is (1, 1). The "Orthogonal projection" status is true.
5. If you set both u1 = 1, u2 = 1 and v1 = 2, v2 = 2, the status reads "Range and direction do not form a basis" because they point in the same direction.