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  2. Projection (linear algebra) - Wikipedia

    en.wikipedia.org/wiki/Projection_(linear_algebra)

    A square matrix is called a projection matrix if it is equal to its square, i.e. if =. [2]: p. 38 A square matrix is called an orthogonal projection matrix if = = for a real matrix, and respectively = = for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .

  3. Vector projection - Wikipedia

    en.wikipedia.org/wiki/Vector_projection

    The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b ⁡ a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .

  4. Orthographic projection - Wikipedia

    en.wikipedia.org/wiki/Orthographic_projection

    Orthographic projection (also orthogonal projection and analemma) [a] is a means of representing three-dimensional objects in two dimensions.Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, [2] resulting in every plane of the scene appearing in affine transformation on the viewing surface.

  5. Projection matrix - Wikipedia

    en.wikipedia.org/wiki/Projection_matrix

    A matrix, has its column space depicted as the green line. The projection of some vector onto the column space of is the vector . From the figure, it is clear that the closest point from the vector onto the column space of , is , and is one where we can draw a line orthogonal to the column space of .

  6. Projection (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Projection_(mathematics)

    The projection of the point C itself is not defined. The projection parallel to a direction D, onto a plane or parallel projection: The image of a point P is the intersection of the plane with the line parallel to D passing through P. See Affine space § Projection for an accurate definition, generalized to any dimension. [citation needed]

  7. Transformation matrix - Wikipedia

    en.wikipedia.org/wiki/Transformation_matrix

    As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation. Parallel projections are also linear transformations and can be represented simply by a matrix. However, perspective projections are not, and to represent these with a matrix, homogeneous coordinates can be ...

  8. Numerical methods for linear least squares - Wikipedia

    en.wikipedia.org/wiki/Numerical_methods_for...

    is exactly a sought for orthogonal projection of onto an image of X (see the picture below and note that as explained in the next section the image of X is just a subspace generated by column vectors of X). A few popular ways to find such a matrix S are described below.

  9. Endomorphism - Wikipedia

    en.wikipedia.org/wiki/Endomorphism

    Orthogonal projection onto a line, m, is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism. In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism.