4×4Matrix

A 4×4 matrix is a square array of numbers arranged in four rows and four columns. Denoted by M or A, its entries are usually written m11, m12, ..., m44. In linear algebra, a 4×4 matrix represents a linear transformation from R4 to R4 and acts by matrix multiplication on column vectors of length four. In computer graphics and robotics, 4×4 matrices commonly appear in the homogeneous coordinate framework, where the fourth component allows encoding of translations and projective effects.

Typically, a 4×4 matrix is viewed in terms of blocks: the upper-left 3×3 block handles linear rotation and scaling, the upper-right 3×1 column encodes translation, and the bottom row is often (0 0 0 1) for affine transformations in homogeneous coordinates. A general 4×4 matrix can also represent projective transforms whose bottom row differs from (0 0 0 1).

Core operations include addition and multiplication, with multiplication corresponding to composing transformations. The determinant determines invertibility; a nonzero determinant implies an inverse exists. The inverse matrix undoes the transformation. Eigenvalues and eigenvectors, defined for square matrices, extend to 4×4 cases and reveal intrinsic properties of the transformation.

Applications are broad. In linear algebra, 4×4 matrices can describe systems of four equations and four unknowns. In 3D computer graphics, they encode model, view, and projection transforms, enabling world coordinates to be mapped to screen coordinates. In robotics and physics, they represent pose transformations and coordinate frame changes.

Implementation notes include storage order; some systems use row-major and others column-major memory layouts. Operations are implemented in numerical libraries and depend on the context, such as real-valued matrices or symbolic arithmetic.