Pseudoinverse Calculator Python

Moore-Penrose Pseudoinverse Formula:

\[ A^+ = V \Sigma^+ U^T \]

where \(\Sigma^+\) has \(1/\sigma_i\) on diagonal for \(\sigma_i > 0\), 0 otherwise, from SVD \(A = U \Sigma V^T\)

Pseudoinverse (A⁺):

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1. What is the Moore-Penrose Pseudoinverse?

The Moore-Penrose pseudoinverse is a generalization of the inverse matrix for singular or rectangular matrices. It provides a least-squares solution to systems of linear equations that may not have exact solutions.

2. How Does the Calculator Work?

The calculator computes the pseudoinverse using Singular Value Decomposition (SVD):

\[ A^+ = V \Sigma^+ U^T \]

Where:

\( A \) — Input matrix (m × n)
\( U \) — Left singular vectors (m × m)
\( \Sigma \) — Diagonal matrix of singular values (m × n)
\( V \) — Right singular vectors (n × n)
\( \Sigma^+ \) — Pseudoinverse of Σ (reciprocal of non-zero σ_i)

Explanation: The SVD decomposes the matrix, then the pseudoinverse is constructed by taking reciprocals of non-zero singular values.

3. Importance of Pseudoinverse

Details: The pseudoinverse is crucial in linear algebra, statistics (linear regression), control theory, and machine learning (particularly in solving systems with no unique solution).

4. Using the Calculator

Tips: Enter your matrix with rows separated by semicolons and elements separated by spaces or commas. For example: "1,2,3;4,5,6" for a 2×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: When is the pseudoinverse equal to the regular inverse?
A: When the matrix is square and non-singular (has non-zero determinant).

Q2: What applications use pseudoinverse?
A: Least squares regression, robotics, signal processing, and solving underdetermined systems.

Q3: How does Python compute pseudoinverse?
A: Typically via numpy.linalg.pinv which uses SVD internally.

Q4: What's the computational complexity?
A: O(min(mn², m²n)) for an m×n matrix, dominated by the SVD computation.

Q5: Can pseudoinverse handle rank-deficient matrices?
A: Yes, it provides the minimal norm solution for underdetermined systems.