Least Squares Solver using SVD and the Pseudoinverse

Compute the least squares solution \(\mathbf{x} = A^{\dagger}\mathbf{b}\) using the Moore-Penrose pseudoinverse¹.
Most robust method for rank-deficient and ill-conditioned matrices².

The SVD-based method uses the pseudoinverse to solve least squares problems:

\[\mathbf{x} = A^{\dagger}\mathbf{b}\]

How it works:

Compute SVD: \(A = U\Sigma V^T\), then form the pseudoinverse:

\[A^{\dagger} = V\Sigma^{\dagger}U^T\]

where \(\Sigma^{\dagger}\) inverts nonzero singular values and transposes.

Advantages:

Handles rank-deficient matrices automatically
Most numerically stable method³
Provides minimum-norm solution for underdetermined systems
Reveals condition number and numerical rank

Rows \(m\): Columns \(n\):

Enter dimensions and click Generate Matrix to create the input grid and b vector.

Display values as: Fractions Decimals

1 — Set Up

Loads a pre-filled 3×2 overdetermined system — ready to compute.

2 — Compute

Solves the least-squares problem via SVD and the pseudoinverse A⁺ = VΣ⁺Uᵀ.

Resets all dimensions, matrix entries, and results.

Footnotes

The Moore-Penrose pseudoinverse \(A^{\dagger}\) is the unique matrix satisfying four conditions: \(AA^{\dagger}A = A\), \(A^{\dagger}AA^{\dagger} = A^{\dagger}\), \((AA^{\dagger})^T = AA^{\dagger}\), and \((A^{\dagger}A)^T = A^{\dagger}A\). It generalizes the matrix inverse to non-square and singular matrices. ↑
A matrix is rank-deficient when its rank is less than \(\min(m, n)\). The SVD approach handles this gracefully by zeroing out contributions from negligible singular values, unlike the normal equations which may fail. ↑
The SVD is considered the gold standard for numerical stability because it avoids forming \(A^T A\), which can square the condition number. The pseudoinverse is computed as \(A^{\dagger} = V\Sigma^{\dagger}U^T\), where \(\Sigma^{\dagger}\) inverts only the nonzero singular values. ↑