Pseudoinverse Calculator

The pseudoinverse, commonly known as the Moore-Penrose inverse, extends the concept of a matrix inverse to non-square or singular matrices where a standard inverse does not exist. For a matrix A, its pseudoinverse A⁺ is calculated using singular value decomposition (SVD). If A has linearly independent columns, the left pseudoinverse formula is expressed as A⁺ = (A^TA)^-1A^T. Conversely, if A has linearly independent rows, the right pseudoinverse formula applies: A⁺ = A^T(AA^T)^-1. This calculator automatically detects the dimensions of your input matrix—whether 2x2, 3x3, or rectangular variants like 3x2 or 2x3—and applies the appropriate matrix factorization algorithms to yield accurate solutions.

Consider a simple 2x2 singular matrix A where row 1 is [1, 2] and row 2 is [2, 4]. Because the second row is a direct multiple of the first, the determinant of A is zero, meaning a traditional inverse cannot be calculated. To find the pseudoinverse A⁺, we first compute the transpose A^T and the product matrix A^TA. Next, we determine the inverse of that product (or apply SVD to handle the rank deficiency safely) and multiply it back by A^T. The resulting pseudoinverse matrix elements yield a standardized approximation that minimizes residual errors for linear least-squares problems associated with this matrix.

When entering custom matrix values into the calculator, double-check your sign placements and decimal entries to prevent rounding errors. Ensure that you correctly specify the matrix dimensions (e.g., 2x3 or 3x2) before reviewing the output table elements like x11, x12, or y33. If you are solving data regression problems, remember that the pseudoinverse directly provides the minimum-norm least-squares solution to your linear system.