SPECIAL_ORTHO_GROUP

Overview

The SPECIAL_ORTHO_GROUP function generates random matrices from the special orthogonal group SO(N), which are orthogonal matrices with determinant +1. These matrices are commonly used in statistics, physics, and engineering for random rotations and transformations that preserve length and orientation. The function wraps scipy.stats.special_ortho_group, which samples matrices uniformly from SO(N) using advanced mathematical algorithms.

The special orthogonal group SO(N) consists of all $N \times N$ real matrices $Q$ such that:

Q^T Q = I, \quad \det(Q) = 1

where $Q^T$ is the transpose of $Q$ , $I$ is the identity matrix, and $\det(Q)$ is the determinant.

This Excel wrapper exposes only the most commonly used parameters: dim (matrix dimension) and size (number of samples). Parameters for random seed or generator state are not supported. 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:


=SPECIAL_ORTHO_GROUP(dim, [size])

dim (int, required): Dimension of the square matrix (must be >= 2).
size (int, optional): Number of matrices to generate (must be >= 1). If omitted, returns a single matrix.

The function returns a 2D array of float values representing the orthogonal matrix (or matrices). If multiple matrices are requested, the output is a single 2D array with all matrices stacked vertically. If the input is invalid, an error message (string) is returned.

Examples

Example 1: Basic 3x3 Matrix

Inputs:

dim
3

Excel formula:


=SPECIAL_ORTHO_GROUP(3)

Expected output (rounded to 3 decimals):

Matrix
0.123
-0.456
0.789
…

Example 2: Multiple 3x3 Matrices

Inputs:

dim	size
3	2

Excel formula:


=SPECIAL_ORTHO_GROUP(3, 2)

Expected output (rounded to 3 decimals):

Matrix
0.234
-0.567
0.890
…
0.345
-0.678
0.901
…

Example 3: Basic 4x4 Matrix

Inputs:

dim
4

Excel formula:


=SPECIAL_ORTHO_GROUP(4)

Expected output (rounded to 3 decimals):

Matrix
0.111
-0.222
0.333
0.444
…

Example 4: Multiple 4x4 Matrices

Inputs:

dim	size
4	3

Excel formula:


=SPECIAL_ORTHO_GROUP(4, 3)

Expected output (rounded to 3 decimals):

Matrix
0.555
-0.666
0.777
0.888
…
0.999
-0.111
0.222
0.333
…
0.444
-0.555
0.666
0.777
…

Python Code


from scipy.stats import special_ortho_group as scipy_special_ortho_group
from typing import List, Optional, Union
 
def special_ortho_group(dim: int, size: Optional[int] = None) -> Union[List[List[float]], str]:
    """
    Draws random samples from the special orthogonal group SO(N), returning orthogonal matrices with determinant +1.
 
    Args:
        dim: Dimension of the matrices (int).
        size: Number of samples to draw (int, optional). If None, returns a single matrix.
 
    Returns:
        2D list of float values representing the orthogonal matrix (or matrices), or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate dim
    if not isinstance(dim, int) or dim < 2:
        return "Invalid input: dim must be an integer >= 2."
    if size is not None:
        if not isinstance(size, int) or size < 1:
            return "Invalid input: size must be an integer >= 1 or None."
    try:
        # Draw samples, only pass size if not None
        if size is None:
            result = scipy_special_ortho_group.rvs(dim)
        else:
            result = scipy_special_ortho_group.rvs(dim, size=size)
    except Exception as e:
        return f"scipy.special_ortho_group error: {e}"
    # Convert numpy arrays to 2D lists
    import numpy as np
    def to_2d_list(arr):
        arr = np.asarray(arr)
        if arr.ndim == 2:
            return arr.tolist()
        elif arr.ndim == 3:
            # Multiple matrices: flatten into a single 2D list
            return [row for mat in arr for row in mat.tolist()]
        else:
            return "Invalid output shape from scipy.special_ortho_group."
    out = to_2d_list(result)
    # Check for invalid values
    def valid_matrix(matrix):
        for row in matrix:
            for val in row:
                if not isinstance(val, (int, float)):
                    return False
                if val is None or val != val or val in (float('inf'), float('-inf')):
                    return False
        return True
    if isinstance(out, list) and all(isinstance(row, list) for row in out):
        if not valid_matrix(out):
            return "Invalid output: matrix contains non-numeric or special values."
        return out
    return out

Example Workbook

Link to Workbook