Singular Value Decomposition (svd) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of SVD on a simple matrix

What is the shape of the matrices U, S, and VT after applying SVD to a 3x2 matrix using scipy.linalg.svd with full_matrices=False?

SciPy

import numpy as np
from scipy.linalg import svd

A = np.array([[1, 2], [3, 4], [5, 6]])
U, S, VT = svd(A, full_matrices=False)
print(U.shape, S.shape, VT.shape)

A (3, 2) (2,) (2, 2)

B (3, 3) (3,) (3, 2)

C (3, 2) (3,) (2, 2)

D (3, 3) (2,) (2, 2)

💡 Hint

Remember that full_matrices=False returns reduced shapes for U and VT.

When full_matrices=False, U has shape (m, k), S has length k, and VT has shape (k, n), where k = min(m, n). For A with shape (3, 2), k=2, so U is (3, 2), S is (2,), and VT is (2, 2).

❓ data_output

intermediate

Reconstruct matrix from SVD components

Given U, S, and VT from SVD of matrix A, which option correctly reconstructs A?

SciPy

import numpy as np
from scipy.linalg import svd

A = np.array([[7, 8], [9, 10], [11, 12]])
U, S, VT = svd(A, full_matrices=False)

# Which code reconstructs A correctly?

A np.dot(U, S) @ VT

B U * S * VT

C U @ np.diag(S) @ VT

D np.diag(U) @ S @ VT

💡 Hint

Recall that S is a 1D array of singular values, so you need to convert it to a diagonal matrix.

The original matrix A can be reconstructed by multiplying U, the diagonal matrix of singular values, and VT: A = U * diag(S) * VT.

Option C tries element-wise multiplication which is invalid for matrix reconstruction. Option C tries dot product with 1D array S which is invalid. Option C tries to diagonalize U which is incorrect.

❓ visualization

advanced

Visualizing singular values

Which code snippet correctly plots the singular values of matrix A using matplotlib?

SciPy

import numpy as np
from scipy.linalg import svd
import matplotlib.pyplot as plt

A = np.array([[3, 1, 1], [-1, 3, 1]])
U, S, VT = svd(A)

# Which code plots singular values S as a bar chart?

A plt.bar(range(len(S)), S); plt.show()

B plt.plot(S); plt.show()

C plt.scatter(range(len(S)), S); plt.show()

D plt.hist(S); plt.show()

💡 Hint

Bar charts are good for showing discrete values like singular values.

Singular values are discrete and best shown with a bar chart. Option A uses plt.bar which is correct. Option A plots a line which is less clear. Option A uses scatter which is less common here. Option A uses histogram which is for distributions, not singular values.

🧠 Conceptual

advanced

Effect of zero singular values

If a matrix has singular values [5, 3, 0, 0], what does the presence of zero singular values indicate about the matrix?

A The matrix is invertible

B The matrix is full rank with rank 4

C The matrix is symmetric

D The matrix is rank deficient and has rank 2

💡 Hint

Zero singular values mean some dimensions collapse to zero.

Zero singular values mean the matrix does not have full rank. The rank equals the number of non-zero singular values. Here, rank is 2, so the matrix is rank deficient.

It is not full rank (B), symmetry (C) is unrelated to singular values, and invertibility (A) requires full rank.

🔧 Debug

expert

Identify error in SVD reconstruction code

What error will this code produce when trying to reconstruct matrix A from its SVD components?

SciPy

import numpy as np
from scipy.linalg import svd

A = np.array([[1, 0], [0, 1]])
U, S, VT = svd(A)
A_reconstructed = U @ S @ VT
print(A_reconstructed)

A TypeError: unsupported operand type(s) for @: 'numpy.ndarray' and 'numpy.ndarray'

B ValueError: shapes (2,2) and (2,) not aligned for matrix multiplication

C No error, prints the original matrix

D NameError: name 'S' is not defined

💡 Hint

Check the shape and type of S before matrix multiplication.

S is a 1D array of singular values, not a 2D matrix. Multiplying U @ S causes a shape mismatch error because S is (2,) and U is (2,2). You must convert S to a diagonal matrix using np.diag(S) before multiplication.