Pseudoinverse Calculator

The Moore-Penrose pseudoinverse A⁺ is a generalization of the matrix inverse that works for any matrix, including singular and non-square matrices. It satisfies four conditions: AA⁺A = A, A⁺AA⁺ = A⁺, (AA⁺)ᵀ = AA⁺, and (A⁺A)ᵀ = A⁺A.

When A is a square invertible matrix (det(A) ≠ 0), the pseudoinverse equals the regular inverse: A⁺ = A⁻¹. The pseudoinverse reduces to the standard inverse in this special case.

The most common method is using SVD: decompose A = UΣVᵀ, then compute A⁺ = VΣ⁺Uᵀ where Σ⁺ is formed by taking the reciprocal of each non-zero singular value and transposing the resulting diagonal matrix.

For a column vector a (m × 1), the pseudoinverse is a⁺ = aᵀ/(‖a‖²) if a ≠ 0, giving a 1 × m row vector. For the zero vector, the pseudoinverse is the zero row vector.

The pseudoinverse is used in least squares problems (x = A⁺b gives the minimum-norm solution), solving underdetermined systems, data fitting, machine learning, and signal processing.

For the least squares problem min‖Ax — b‖, the solution is x = A⁺b. When A has full column rank, A⁺ = (AᵀA)⁻¹Aᵀ. This gives the best approximate solution when no exact solution exists.

Yes, the Moore-Penrose pseudoinverse is uniquely defined for every matrix. There is exactly one matrix A⁺ that satisfies all four Moore-Penrose conditions.

The pseudoinverse of the transpose satisfies (Aᵀ)⁺ = (A⁺)ᵀ. This property allows you to compute the pseudoinverse of a transpose without performing a separate decomposition.

In general, (AB)⁺ ≠ B⁺A⁺. However, (AB)⁺ = B⁺A⁺ holds when A has full column rank and B has full row rank. The product rule for pseudoinverses is more complex than for regular inverses.

For an orthogonal matrix Q, the pseudoinverse equals the transpose: Q⁺ = Qᵀ. This is because QᵀQ = I, satisfying the conditions for the regular inverse.

For a diagonal matrix D, the pseudoinverse is formed by taking the reciprocal of each non-zero diagonal element and leaving zeros as zeros. For example, diag(2, 0, 3)⁺ = diag(1/2, 0, 1/3).

Computing the pseudoinverse via SVD requires O(m × n × min(m, n)) operations. Using the normal equations (AᵀA)⁻¹Aᵀ requires O(m × n² + n³) operations but requires A to have full column rank.

Yes, the pseudoinverse is specifically designed for non-square matrices. An m × n matrix has an n × m pseudoinverse, which is the generalization that the regular inverse cannot provide.

For a rank-1 matrix uvᵀ, the pseudoinverse is vuᵀ/(‖u‖²‖v‖²). This follows from the SVD of rank-1 matrices where the single singular value is ‖u‖‖v‖.

The four conditions are: (1) AA⁺A = A, (2) A⁺AA⁺ = A⁺, (3) (AA⁺)ᵀ = AA⁺ (hermitian symmetry), and (4) (A⁺A)ᵀ = A⁺A (hermitian symmetry). These uniquely define the pseudoinverse.

The pseudoinverse is computed directly from SVD: if A = UΣVᵀ, then A⁺ = VΣ⁺Uᵀ. The SVD provides the most numerically stable way to compute the pseudoinverse.

The pseudoinverse of a zero matrix of size m × n is the zero matrix of size n × m. Since there are no non-zero singular values, Σ⁺ contains only zeros.

For underdetermined systems Ax = b, x = A⁺b gives the solution with minimum Euclidean norm ‖x‖. This is the "shortest" solution among all possible solutions.

Verify the four Moore-Penrose conditions: (1) AA⁺A = A, (2) A⁺AA⁺ = A⁺, (3) AA⁺ is symmetric, (4) A⁺A is symmetric. All four must hold for a correct pseudoinverse.

Yes, pseudoinverse is fundamental in linear regression (normal equations), principal component analysis, recommender systems, and neural network weight initialization. It provides stable solutions to ill-conditioned problems.

The regular inverse only exists for square invertible matrices. The pseudoinverse works for any matrix — square or rectangular, invertible or singular — making it far more general. For invertible square matrices, they are identical.

The Pseudoinverse Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Pseudoinverse Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.