Ickesingulär

Ickesingulär (non-singular) is a term used in linear algebra to denote a square matrix that is invertible; equivalently, its determinant is nonzero. Invertible matrices have full rank and, in particular, their columns are linearly independent.

Several statements are equivalent to being ickesingulär: the matrix has full rank equal to its size; there exists a unique inverse B such that AB = BA = I; the associated linear transformation is an isomorphism, hence bijective from the vector space to itself; and all eigenvalues are nonzero.

Consequences of ickesingulär include that the inverse exists and is unique, the kernel is {0}, and the image covers the entire space. Consequently, the system Ax = b has a unique solution for every right-hand side b.

Verification methods include computing the determinant and confirming it is nonzero, determining the rank and checking that it equals the matrix size, or attempting to compute the inverse. In numerical work, LU decomposition or Gaussian elimination with nonzero pivots are common tools.

Example: A = [[1, 2], [3, 5]] has det(A) = 1×5 − 2×3 = −1 ≠ 0, so it is ickesingulär, and its inverse is [[-5, 2], [3, -1]]. By contrast, B = [[1, 2], [2, 4]] has det(B) = 0 and is singular, hence not ickesingulär.