Pseudo-Inverse

Shipping

matrix_pinv

Moore-Penrose pseudo-inverse A⁺ for rectangular / rank-deficient matrices

Signature

Inputs

aSignal|Arrayrequired— A 2-D matrix (any shape, possibly rank-deficient).

Outputs

pseudo_inverseMatrix— The Moore-Penrose pseudo-inverse $A^{+}$.

Description

Pseudo-Inverse computes the Moore-Penrose pseudo-inverse $A^{+}$ via the MKL LAPACK SVD, generalizing the matrix inverse to rectangular and rank-deficient matrices. For a full-rank square matrix $A^{+} = A^{- 1}$ ; otherwise $A^{+}$ gives the minimum-norm least-squares solution operator $x = A^{+} b$ .

The input may be any 2-D matrix. The result is dimensionless. When you actually need to solve $A x \approx b$ rather than obtain the operator itself, matrix_lstsq is usually the better node; use pseudo-inverse when the $A^{+}$ matrix is the desired quantity (e.g. a reusable projection).

Mathematics

A^{+} = V Σ^{+} U^{⊤}, Σ_{ii}^{+} = {1/ σ_{i} 0 σ_{i} > tol else

x = A^{+} b = ar g x min ∥ A x - b ∥_{2} (minimum norm)

Examples

Inverting a non-square map

For a wide $2 \times 3$ matrix $A$ , no ordinary inverse exists, but pseudo_inverse returns a $3 \times 2$ matrix $A^{+}$ such that $A A^{+} = I_{2}$ . Applying $A^{+}$ to an observation yields the minimum-norm solution of the under-determined system.

Applications

Least-squares operators for over/under-determined and rank-deficient systems.
Robotics redundancy resolution (minimum-norm joint velocities via the Jacobian pseudo-inverse).
Projection and reconstruction in signal processing.
Regularized inversion where a true inverse is undefined.

Neat

Built on the SVD, it thresholds tiny singular values to zero, so rank-deficient matrices invert gracefully instead of blowing up.

For a full-rank square matrix it coincides exactly with matrix_inverse, making it a safe superset.

Known issues

The singular-value truncation tolerance affects results for near-rank-deficient matrices; borderline singular values can flip in or out.

To just solve a system, matrix_lstsq is cheaper and reports residual/rank — pinv materializes the whole operator.