squarematrix

A squarematrix, also called a square matrix, is a matrix with the same number of rows and columns. In standard notation, a square matrix of size n is written as A ∈ R^{n×n} (or more generally over any field), with entries a_{ij} for i, j = 1,..., n. The transpose A^T, the determinant det(A), and the identity matrix I_n are defined for square matrices.

Operations and basic properties: Square matrices can be added and multiplied by compatible matrices, and multiplied by scalars. The determinant det(A) provides information about invertibility; det(A) ≠ 0 means A is nonsingular and has an inverse A^{-1}. If det(A) = 0, A is singular and may not have an inverse.

Eigenstructure and trace: For a square matrix A, there exist scalars λ called eigenvalues and nonzero vectors v such that A v = λ v. They are found from det(A - λ I_n) = 0. The trace tr(A) is the sum of diagonal entries a_{ii}.

Special cases: Symmetric A = A^T, skew-symmetric A = -A^T, diagonal, triangular; orthogonal matrices satisfy A^T A = I_n, and positive definite matrices satisfy x^T A x > 0 for all nonzero x. Square matrices may be diagonalized (A = P D P^{-1}) when they are diagonalizable, with D diagonal containing eigenvalues.

Applications: Square matrices model linear transformations, solve linear systems via Gaussian elimination, power iteration for eigenvalues, and are fundamental in computer graphics, physics, and data analysis.