PINV

Overview

The PINV function computes the Moore-Penrose pseudoinverse of a matrix using singular value decomposition (SVD). The pseudoinverse is a generalization of the matrix inverse that exists for any matrix, including non-square or singular matrices. It is widely used to solve linear systems that may not have a unique solution, and in applications such as least squares fitting and signal processing. The pseudoinverse $A^+$ of a matrix $A$ is the unique matrix that satisfies the four Moore-Penrose conditions:

where $A^*$ denotes the conjugate transpose. If $A$ is invertible then the Moore-Penrose pseudoinverse is exactly the inverse of $A$. If $A$ is not invertible then the Moore-Penrose pseudoinverse computes the $x$ solution to $Ax = b$ such that $\|Ax - b\|$ is minimized. For more details, see the SciPy documentation .

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:

=PINV(matrix, [atol], [rtol], [return_rank], [check_finite])

• matrix (2D list, required): The matrix to be pseudo-inverted. Must be a 2D list of numbers.
• atol (float, optional, default=0.0): Absolute threshold for small singular values to be considered zero.
• rtol (float, optional, default=None): Relative threshold for small singular values to be considered zero. If None, uses max(M, N) * eps.
• return_rank (bool, optional, default=False): If True, also returns the effective rank of the matrix.
• check_finite (bool, optional, default=True): If True, checks that the input contains only finite numbers.

The function returns the pseudoinverse of the matrix as a 2D list. If return_rank is True, returns a list [pseudoinverse, rank]. If the input is invalid, returns an error message (string or 2D list of strings).

Wrap all JSX or HTML characters outside code blocks in backticks.

Examples

Example 1: Pseudoinverse of a 2x2 invertible matrix

Inputs:

Excel formula:

=PINV({1,2;3,4})

Expected output:

Example 2: Pseudoinverse of a 3x2 matrix

Inputs:

Excel formula:

=PINV({1,2;3,4;5,6})

Expected output:

Example 3: Pseudoinverse with return_rank=TRUE

Excel formula:

=PINV({1,2;3,4}, , , TRUE)

Expected output:

Example 4: Pseudoinverse with atol and rtol

Excel formula:

=PINV({1,2;3,4}, 1e-5, 1e-5)

Expected output:

Python Code

import numpy as np
from scipy.linalg import pinv as scipy_pinv
 
def pinv(matrix, atol=0.0, rtol=None, return_rank=False, check_finite=True):
    """
    Compute the Moore-Penrose pseudoinverse of a matrix using SVD.
 
    Args:
        matrix: 2D list of numbers to be pseudo-inverted.
        atol: Absolute threshold for small singular values (default: 0.0).
        rtol: Relative threshold for small singular values (default: None).
        return_rank: If True, only return the effective rank as a 2D list (default: False).
        check_finite: If True, check that the input contains only finite numbers (default: True).
 
    Returns:
        2D list pseudoinverse of the matrix, or [[rank]] if return_rank is True, or error message (str or 2D list of str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate input
    if not isinstance(matrix, list) or not matrix or not isinstance(matrix[0], list):
        return [["Invalid input: matrix must be a 2D list."]]
    try:
        arr = np.array(matrix, dtype=float)
    except Exception:
        return [["Invalid input: matrix must contain only numbers."]]
    try:
        if return_rank:
            pinv_result, rank = scipy_pinv(arr, atol=atol, rtol=rtol, return_rank=True, check_finite=check_finite)
            return [[int(rank)]]
        else:
            result = scipy_pinv(arr, atol=atol, rtol=rtol, check_finite=check_finite)
            return result.tolist()
    except Exception as e:
        return [[f"scipy.linalg.pinv error: {e}"]]

Live Notebook

Edit this function in a live notebook .

Live Demo