noninvertibility
Noninvertibility is the property of lacking an inverse. In mathematics, an inverse function f^{-1} or a matrix A^{-1} does not exist when the operation is not bijective or the transformation collapses information. In physics and other fields, noninvertibility means the original state cannot be uniquely recovered from its image.
In linear algebra, a square matrix A is noninvertible if det(A) = 0; equivalently, rank(A) < n. Such
Noninvertibility also occurs for non-square matrices. A matrix with full row rank has a left inverse, and
For practical purposes, the Moore–Penrose pseudoinverse A+ provides a best approximate inverse for noninvertible matrices, and
In dynamics and information theory, noninvertible maps cause information loss: repeated application can merge distinct states,
Examples include the linear map f(x, y) = (x, 0) represented by the matrix [[1, 0], [0, 0]]