Matrixsysteme

Matrixsysteme (systems of linear equations) are collections of linear equations that share a common set of unknowns. A typical system with n variables x1, x2, ..., xn consists of m equations: a11 x1 + a12 x2 + ... + a1n xn = b1, ..., am1 x1 + am2 x2 + ... + amn xn = bm.

These systems can be written in matrix form as A x = b, where A is an m-by-n coefficient matrix, x is the n-dimensional column vector of variables, and b is the right-hand side vector. The study of Matrixsysteme focuses on existence and characterization of solutions, using concepts such as the rank of A and of the augmented matrix [A|b].

Solvability is determined by the rank conditions. A system is consistent if rank([A|b]) = rank(A). It has a unique solution when rank(A) = rank([A|b]) = n (the number of variables). If rank(A) = rank([A|b]) = r < n, the system has infinitely many solutions. If rank([A|b]) > rank(A), the system is inconsistent and has no solution. For square systems (m = n), a unique solution exists when det(A) ≠ 0.

Common methods for solving Matrixsysteme include Gaussian elimination (row reduction to reduced row echelon form), LU decomposition, and, in the square non-singular case, using the matrix inverse or Cramer's rule. Large or sparse systems are often approached with iterative methods such as Jacobi or Gauss-Seidel.

Applications span engineering, physics, computer graphics, economics, and statistics. The concept provides a foundational framework for linear modeling, optimization, and numerical analysis.