Matrix theory

Matrix theory studies matrices and their algebraic, geometric, and computational properties, including how matrix operations encode linear transformations and systems of equations.

Matrix: A rectangular array of numbers or other elements arranged in rows and columns. Matrices are the objects whose operations and properties matrix theory studies.
Gaussian elimination: A sequence of row operations that reduces a linear system or matrix to a simpler form. It solves systems and determines rank using elementary matrix operations.
System of linear equations: A collection of linear equations sharing the same unknowns. Writing coefficients as a matrix turns solving the system into a central matrix problem.
Diagonalizable matrix: A square matrix similar to a diagonal matrix. Diagonalization simplifies powers and functions when enough independent eigenvectors exist.
Matrix multiplication algorithm: An algorithm for computing the product of two matrices. The best possible exponent for multiplying square matrices remains unknown.
Matrix multiplication: An operation combining compatible matrices by taking row-by-column products and sums. It composes linear transformations and drives much of matrix algebra.
LU decomposition: A factorization of a matrix into a lower-triangular matrix and an upper-triangular matrix. It turns repeated linear-system solves into efficient triangular substitutions.
Least squares: A method that chooses parameters to minimize the sum of squared residuals. Matrix factorizations solve overdetermined data-fitting problems efficiently.
Nilpotent matrix: A square matrix whose power is zero for some positive integer exponent. It contrasts with invertible behavior and exposes structure hidden from eigenvalues alone.
Numerical stability: The study of how rounding errors affect the accuracy of numerical algorithms. Matrix computations can amplify tiny errors, making stability as important as speed.

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