n×nmatriisin

An n×n matrix, or square matrix of order n, is a rectangular array containing n rows and n columns of elements, usually denoted aij for i and j from 1 to n. When the entries come from a field F (such as the real numbers R or the complex numbers C), the matrix is written as A ∈ Fn×n. Matrices serve as compact representations of linear transformations from an n-dimensional vector space to itself: applying A to a vector x produces Ax.

Common operations include matrix addition and scalar multiplication, matrix multiplication, and transposition. The product of two n×n matrices corresponds to the composition of the associated linear transformations. The determinant det(A) is a scalar function that vanishes precisely when A is singular (non-invertible). A matrix is invertible if det(A) ≠ 0, in which case it has a unique inverse A−1 satisfying AA−1 = A−1A = I.

Special forms include diagonal matrices (only diagonal entries nonzero), triangular matrices (upper or lower), symmetric matrices (Aᵀ = A), and, over C, unitary matrices (AᵀA = I or ĀᵀA = I). Eigenvalues λ satisfy det(A − λI) = 0, and eigenvectors are nonzero vectors v with Av = λv.

The rank of an n×n matrix is the dimension of its column (or row) space; full rank means rank n and implies invertibility. Applications span solving linear systems Ax = b, coordinate changes, computer graphics, statistics, and machine learning. Computational methods include LU decomposition for solving systems and computing determinants, and eigenvalue decompositions for spectral properties. In Finnish terminology, this object is commonly called an n×n matriisi.