The SVD of a matrix can be calculated by using the svd function within scipy.linalg.

Let us understand how to code up using the svd function using the following example:

1. Initialize an array:

 a = np.array([[1, 2], [3, 4]]) 

2. Import the relevant packages and functions within the package:

from scipy import linalg 

3. Use the svd function to calculate the SVD matrices (U, Σ and V):

linalg.svd(a)

4. The output of the preceding function is as follows:

(array([[-0.40455358, -0.9145143 ],
        [-0.9145143 ,  0.40455358]]),
 array([ 5.4649857 ,  0.36596619]),
 array([[-0.57604844, -0.81741556],
        [ 0.81741556, -0.57604844]]))

​In order to further understand if the output is as per expectation, let us check if the multiplication of the individual matrices in the output results in a matrix that is very similar to our original matrix. 

In this scenario, our U matrix is as follows:

([[-0.40455358,-0.9145143],[-0.9145143,0.40455358]])

The Σ matrix is as follows:

([ 5.4649857 ,  0.36596619])

However, note that the preceding second output of the svd function is only the diagonal elements within the matrix (where every non-diagonal element is 0). Hence, the Σ matrix translates to the following:

np.array([[ 5.4649857 ,  0],
                  [0,0.36596619]])

Note that the diagonal values are populated by the elements of the svd output we saw earlier.

The V matrix is as follows:

([[-0.57604844, -0.81741556],  [ 0.81741556, -0.57604844]])

In order to cross-check and verify the svd function's output, we will multiply the U, Σ, and V matrices that we obtained earlier.

The output of the matrix multiplication should be as close to the original matrix as possible.

1. Initialize the matrices:

U = np.array([[-0.40455358, -0.9145143 ],
        [-0.9145143 ,  0.40455358]])
sigma = np.array([[ 5.4649857 ,  0],
                  [0,0.36596619]])
V=np.array([[-0.57604844, -0.81741556],
        [ 0.81741556, -0.57604844]])

2. Perform the matrix multiplication of U, sigma, and V matrices:

np.dot(np.dot(U,sigma),V)

3. The output of the preceding calculation is as follows:

array([[ 0.99999999,  1.99999998],
       [ 3.00000003,  4.00000001]])

 Note that, the output matrix is almost the same as the original matrix.