NxN

NxN refers to a square matrix or array with N rows and N columns. It represents a linear map from an N-dimensional vector space to itself and is a fundamental object in linear algebra. The entries aij come from a field, typically the real or complex numbers. The size is described as N by N.

Operations on NxN matrices include addition, subtraction, and multiplication, as well as scalar multiplication. The transpose A^T reshapes the matrix by swapping rows and columns. The determinant det(A) is defined for square matrices and is zero exactly when A is singular, meaning it does not have full rank. If det(A) ≠ 0, the matrix is invertible and there exists an inverse A^{-1} such that A A^{-1} = I_N, where I_N is the identity matrix with ones on the diagonal and zeros elsewhere. Eigenvalues λ and eigenvectors v satisfy A v = λ v. The determinant equals the product of eigenvalues (with multiplicity) and the trace equals the sum of eigenvalues.

Common matrices include the identity matrix I_N, the zero matrix 0_N, diagonal matrices diag(d1,...,dN), and the all-ones matrix J_N. Permutation matrices effect basis changes and have a single 1 in each row and column.

NxN matrices are used to solve systems of linear equations, to represent linear transformations from R^N to R^N, and across fields such as computer graphics, statistics, and data science. In computation, dense NxN matrices require O(N^2) storage, while sparse matrices store only nonzero entries, enabling efficiency for large N. For small N, explicit examples illustrate concepts like determinants and inverses.