SciPy - expm() Function

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The matrix exponential of a square matrix is calculated in SciPy using the expm(A) method. In differential equations and control theory, for example, this technique is essential for modeling systems that change over time. It offers a method for determining a matrix's exponential utilizing effective numerical techniques.

Even for non-diagonalizable matrices, expm(A) is effective, yielding reliable and precise results for a variety of uses. One of the most important methods for modeling systems in physics, economics, and engineering, it is essential for resolving time-dependent issues.

It is integrated with like scipy.integrate.solve_ivp for solving equations and controls theory for modeling time evaluation and system dynamics.

Syntax

Following is the syntax of the SciPy expm() method

.expm(A)

Parameters

This method accepts the following parameters −

• A − A sqaure matrix of shape(n,n).

This method in SciPy doesn't accept any additional parameters directly. It is designed for the computation of the matrix exponential of a sqaure matrix A.

Return Value

The expm(A) returns a NumPy array, where it represents the matrix exponential of the input square matrix A.

Example 1

This is the basic example of expm() method demonstrates the simple use case when we compute matrix exponential of a 2x2 matrix.

In this example, we define a 2x2 matrix and use the `scipy.linalg.expm` function to compute its matrix exponential. After that, the outcomethe matrix exponential of `A`is printed.

from scipy.linalg import expm
import numpy as np

# Define a 2x2 matrix
A = np.array([[0, 1],
              [-1, 0]])

# Compute the matrix exponential
result = expm(A)

#print the result
print("Matrix A:")
print(A)
print("\nExponential of A (e^A):")
print(result)

When we run above program, it produces following result

Matrix A:
[[ 0  1]
 [-1  0]]

Exponential of A (e^A):
[[ 0.54030231  0.84147098]
 [-0.84147098  0.54030231]]

Example 2: Matrix Exponential of a Larger Matrix

This example demonstrates the use of the matrix exponential to solve a larger matrix.

In this example, we define a 3x3 matrix B and calculate its matrix exponential using the function scipy.linalg.expm. We then print the result which is the matrix exponential of B.

import numpy as np
from scipy.linalg import expm

# Define the matrix B
B = np.array([[1, 2, 3], [0, 1, 4], [5, 6, 0]])

# Calculate the matrix exponential of B
exp_B = expm(B)

print("Matrix exponential of B:\n", exp_B)
  

When we run above program, it produces following result

Matrix exponential of B:
 [[333.93090511 506.07389874 416.91659677]
 [311.08568303 473.91946248 389.97643141]
 [487.47053927 740.50748863 609.7396169 ]]

Example 3: Matrix Exponential of a Diagonal matrix

This example demonstrates the use of the matrix exponential to solve a diagonal matrix.

In this example, we build a diagonal matrix `D` with diagonal elements `[1, 2, 3]` and use the `scipy.linalg.expm` function to compute its matrix exponential. The matrix exponential of `D`: is the result.

import numpy as np
from scipy.linalg import expm

# Define a diagonal matrix
D = np.diag([1, 2, 3])

# Calculate the matrix exponential of the diagonal matrix
exp_D = expm(D)

print("Matrix Exponential of D:\n", exp_D) 

Following is an output of the above code

Matrix Exponential of D:
 [[ 2.71828183  0.          0.        ]
 [ 0.          7.3890561  0.        ]
 [ 0.          0.         20.08553692]]