EXPM

Overview

The EXPM function computes the matrix exponential of a square matrix. The matrix exponential is a fundamental operation in linear algebra, with applications in solving systems of linear differential equations, quantum mechanics, control theory, and more. The matrix exponential of a square matrix $A$ is defined as:

\exp(A) = \sum_{k=0}^{\infty} \frac{A^k}{k!}

This function uses the algorithm from Awad H. Al-Mohy and Nicholas J. Higham (2009), which is a scaling and squaring method with a variable-order Padé approximation, as implemented in scipy.linalg.expm . 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:


=EXPM(matrix)

matrix (2D list, required): A square matrix (n x n) of real or complex numbers.

The function returns a 2D list (n x n) representing the matrix exponential of the input matrix, or a 2D list of strings with an error message if the input is invalid.

Examples

Example 1: Exponential of the Zero Matrix

In Excel:


=EXPM({0,0;0,0})

Expected output:


1.0	0.0
0.0	1.0

Python Code


import numpy as np
from scipy.linalg import expm as scipy_expm
 
def expm(matrix):
    """
    Compute the matrix exponential of a square matrix.
 
    Args:
        matrix: 2D list (n x n) of real or complex numbers.
 
    Returns:
        2D list (n x n) representing the matrix exponential, or a 2D list of strings with an error message if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate input is a 2D list
    if not isinstance(matrix, list) or not matrix or not isinstance(matrix[0], list):
        return [["Invalid input: matrix must be a 2D list."]]
    n = len(matrix)
    if any(len(row) != n for row in matrix):
        return [["Invalid input: matrix must be square."]]
    try:
        arr = np.array(matrix, dtype=np.complex128)
    except Exception:
        return [["Invalid input: matrix must contain only numbers."]]
    try:
        result = scipy_expm(arr)
    except Exception as e:
        return [[f"scipy.linalg.expm error: {e}"]]
    # Convert result to 2D list, round real and imag parts for display
    out = []
    for row in result:
        out_row = []
        for val in row:
            if np.iscomplex(val):
                if abs(val.imag) < 1e-10:
                    out_row.append(float(np.round(val.real, 4)))
                elif abs(val.real) < 1e-10:
                    out_row.append(f"{np.round(val.imag, 4)}j")
                else:
                    out_row.append(f"{np.round(val.real, 4)}+{np.round(val.imag, 4)}j")
            else:
                out_row.append(float(np.round(val, 4)))
        out.append(out_row)
    return out

Live Demo

Example Workbook

Link to Workbook