NxN0

NxN0 denotes an N by N0 matrix, a rectangular array with N rows and N0 columns. The standard notation is A ∈ F^{N×N0}, where F is a field such as the real numbers, and a_{ij} denotes the entry in row i and column j (i = 1,...,N; j = 1,...,N0).

This matrix represents a linear transformation from F^{N0} to F^N: x ∈ F^{N0} maps to Ax ∈ F^N. The rank of A is the dimension of its column space, with rank(A) ≤ min(N, N0). If N = N0, A is square and may be invertible when rank = N.

Operations on NxN0 matrices follow standard linear algebra. The transpose A^T is an N0×N matrix. Multiplication is defined as AB where B ∈ F^{N0×P}, producing AB ∈ F^{N×P}. Row operations can transform A into echelon or reduced row echelon form, and the rank equals the number of pivots in any such form.

Example: A = [ [1, 2], [3, 4], [5, 6] ] is a 3×2 matrix. If B = [ [7,8,9], [10,11,12] ] is 2×3, then AB = [ [27, 30, 33], [61, 68, 75], [95, 106, 117] ].

Applications: NxN0 matrices are ubiquitous in linear systems, data representation (N samples by N0 features), and linear transformations across the sciences. They underpin decompositions such as LU and QR for rectangular matrices, as well as singular value decomposition for general cases. The properties depend on N, N0, and the underlying field F.