In this tutorial, we are going to learn how to raise a matrix to a given power in linear algebra by using the linalg.matrix_power method in Python present in the NumPy module.
The numpy.linalg.matrix_power() method is used to raise a square matrix to an integer power n.
Let us see the syntax of the function first.
Also check: Numpy linalg.eig – Compute the eigenvalues and right eigenvectors of a square array
Syntax of numpy.linalg.matrix_power
numpy.linalg.matrix_power(a, n)
- Parameters:
- a, an MxM array. Input matrix to be raised to a power.
- n, integer, the power or exponent. It can be positive, negative or zero.
- Returns: a**n. The returned matrix is of the same shape as a. If n is positive or zero, then the return type is integer. If n is negative, then the return type is float.
- Raises: LinAlgError for matrices that are not square or (for negative powers) whose inverse cannot be calculated.
Note: If the determinant of a matrix is zero, then its inverse cannot be calculated.
This function is very similar to the numpy.power(n, p) function which takes two parameters, a number n and a power p and raises n to the power p.
Examples of numpy.linalg.matrix_power
Let’s now start with a few examples of the numpy linalg matrix power method.
Using numpy.linalg.matrix_power with positive power
import numpy as np
matrix = [[2, 5], [1, 3]]
# calculating the matrix power
mat_power_2 = np.linalg.matrix_power(matrix, 2)
mat_power_3 = np.linalg.matrix_power(matrix, 3)
print("Matrix = \n", matrix,
"\nMatrix power 2 = \n", mat_power_2,
"\nMatrix power 3 = \n", mat_power_3)
Output:
Matrix =
[[2, 5], [1, 3]]
Matrix power 2 =
[[ 9 25]
[ 5 14]]
Matrix power 3 =
[[ 43 120]
[ 24 67]]
Matrix raised to power 2 is calculated by multiplying the matrix by itself as follows:
The above matrix is the result of matrix power 2. Now, to calculate matrix power 3, we can multiply matrix power 2 by the given matrix. That is,
Using numpy.linalg.power() with negative power
When we pass a negative power, n, to the function, it first calculates the inverse of the matrix and then raises the inverse to the power abs(n).
For a 2×2 matrix like:
The inverse is calculated as:
import numpy as np
matrix = [[2, 5], [1, 3]]
# calculating the matrix power
mat_power = np.linalg.matrix_power(matrix, -2)
print("Matrix = \n", matrix, "\nMatrix power -2 = \n", mat_power)
Output:
Matrix =
[[2, 5], [1, 3]]
Matrix power -2 =
[[ 14. -25.]
[ -5. 9.]]
In this example,
Its inverse is calculated as follows:
Now, raise this inverse of the matrix to the power abs(-2) i.e. 2 as shown below:
Using numpy.linalg.matrix_power with 0
When zero is passed as the power to the numpy.linalg.matrix_power function, an identity matrix of the same shape as the input matrix is returned.
import numpy as np
matrix_1 = [[2, 5], [1, 3]]
matrix_2 = [[4, 2, 5], [1, 8, 3], [6, 0, 2]]
# calculating the matrix power
mat_1_power_0 = np.linalg.matrix_power(matrix_1, 0)
mat_2_power_0 = np.linalg.matrix_power(matrix_2, 0)
print("Matrix 1 = \n", matrix_1,
"\nMatrix 1 power 0 = \n", mat_1_power_0,
"\nMatrix 2 = \n", matrix_2,
"\nMatrix 2 power 0 = \n", mat_2_power_0)
Output:
Matrix 1 =
[[2, 5], [1, 3]]
Matrix 1 power 0 =
[[1 0]
[0 1]]
Matrix 2 =
[[4, 2, 5], [1, 8, 3], [6, 0, 2]]
Matrix 2 power 0 =
[[1 0 0]
[0 1 0]
[0 0 1]]
As matrix 1 is a 2×2 matrix, the output is a 2×2 identity matrix and similarly, the output of matrix 2 raised to 0 is a 3×3 identity matrix.
Conclusion
So, in this tutorial, we learned about numpy.linalg.matrix_power function which is used in linear algebra to compute the power of a square matrix. We also saw various examples of the possible inputs and outputs.
If you want to learn more about NumPy, feel free to go through our NumPy tutorials.