Ruutmatrikside

Ruutmatrikside is the plural term for square matrices, i.e., matrices with the same number of rows and columns. They are central objects in linear algebra and can represent linear transformations from an n-dimensional vector space to itself when defined over a field F (such as the real numbers R or complex numbers).

A square matrix A is called n-by-n if its dimensions are n × n, often written A ∈ F^{n×n}. The set of all n-by-n matrices over F forms an algebra under addition and multiplication, includes the identity I_n and the zero matrix 0.

Key properties: determinant det(A) tells whether A is invertible; invertible if det(A) ≠ 0, with inverse A^{-1}. The trace tr(A) is the sum of diagonal entries. The eigenvalues λ satisfy det(A − λI) = 0; eigenvectors define eigenbasis when A is diagonalizable.

Special classes: diagonal matrices, symmetric matrices A^T = A, orthogonal matrices A^T A = I, triangular matrices. Matrix A represents a linear transformation; its effect can be studied via eigen-decomposition A = PDP^{-1} (when possible) or Jordan form.

Applications and computation: square matrices model systems of linear equations, computer graphics transformations, network analysis, and more. Numerical methods compute determinants, inverses, eigenvalues, and factorizations; complexity grows with n.