1. What is the Pseudoinverse?

The pseudoinverse (or Moore-Penrose inverse) is a generalization of the matrix inverse that can be applied to any m × n matrix. It is particularly useful when dealing with matrices that are not square or are rank-deficient.

2. How Does the Calculator Work?

The calculator uses the SVD-based formula:

Where:

• \( A = U \Sigma V^T \) is the Singular Value Decomposition (SVD)
• \( \Sigma^+ \) has \(1/\sigma_i\) on diagonal for \( \sigma_i > 0 \), 0 otherwise
• \( U, V \) are orthogonal matrices from SVD

Explanation: The pseudoinverse provides the least-squares solution to a system of linear equations when no exact solution exists.

3. Importance of Pseudoinverse

Details: The pseudoinverse is crucial in many applications including linear regression, control theory, signal processing, and solving systems of linear equations when the matrix is not invertible.

4. Using the Calculator

Tips: Enter your matrix using comma to separate values within a row and semicolon to separate rows. For example: "1,2,3;4,5,6" represents 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 full rank), the pseudoinverse is identical to the regular inverse.

Q2: What are the properties of the pseudoinverse?
A: The pseudoinverse satisfies four Moore-Penrose conditions: AA⁺A = A, A⁺AA⁺ = A⁺, (AA⁺)^T = AA⁺, (A⁺A)^T = A⁺A.

Q3: How is pseudoinverse used in linear regression?
A: For the system Xβ = y, the least-squares solution is β = X⁺y, where X⁺ is the pseudoinverse of X.

Q4: Are there computational alternatives to SVD for pseudoinverse?
A: Yes, other methods include QR decomposition or iterative methods, but SVD is the most numerically stable approach.

Q5: What happens when a matrix has full column rank?
A: For matrix A with full column rank, A⁺ = (A^TA)^-1A^T.