4x4n

4x4n is a dimension specification used in linear algebra to denote a matrix with four rows and 4n columns, where n is a positive integer. Formally, a 4x4n matrix M ∈ R^{4×4n} maps vectors from R^{4n} to R^4. The total number of entries is 16n. The row space is a subspace of R^{4n}, while the column space is a subspace of R^4.

Rank and inverses: The rank of M is at most min(4, 4n) = 4. If M has full row rank (rank 4), the rows are independent and M has a left inverse B of size 4n × 4 such that B M = I_4, e.g., B = M^T (M M^T)^{-1}. This reflects that M can be used to uniquely reconstruct any four-dimensional output from a suitable left-inverse perspective.

Structure and construction: In practice, a 4x4n matrix can be formed by horizontally concatenating blocks to reach 4n columns or by arranging arbitrary column vectors in R^4. It often appears in problems with four equations and a higher-dimensional set of unknowns, or in designs where four-output measurements are taken from a high-dimensional input.

Example: For n=2, M is 4×8; for instance M = [I4 0], where I4 is the 4×4 identity matrix and 0 is a 4×4 zero block. This matrix has rank 4 and maps R^8 to R^4.

Applications: Such matrices commonly represent linear relations from a high-dimensional input to four outputs, appear in under-determined systems, and feature in block-matrix or measurement-design contexts.

See also: Matrix, Rectangular matrix, Rank, Left inverse, Pseudoinverse.