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Quiz: Matrix Decompositions

Test your understanding of SVD, QR, and other matrix decompositions.

1. The Singular Value Decomposition (SVD) factors a matrix \(A\) as:

  1. \(A = U\Sigma V\)
  2. \(A = U\Sigma V^T\)
  3. \(A = \Sigma UV\)
  4. \(A = U + \Sigma + V\)
Show Answer

The correct answer is B. The SVD decomposes any \(m \times n\) matrix as \(A = U\Sigma V^T\), where \(U\) and \(V\) are orthogonal matrices and \(\Sigma\) is a diagonal matrix of singular values.

Concept Tested: Singular Value Decomposition

2. Singular values of a matrix are always:

  1. Complex numbers
  2. Negative or zero
  3. Non-negative real numbers
  4. Equal to the eigenvalues
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The correct answer is C. Singular values are always non-negative real numbers, typically arranged in decreasing order: \(\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_r > 0\). They are the square roots of eigenvalues of \(A^TA\).

Concept Tested: Singular Values

3. The rank of a matrix equals:

  1. The number of rows
  2. The number of non-zero singular values
  3. The largest singular value
  4. The trace
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The correct answer is B. The rank of a matrix equals the number of non-zero singular values. This provides a robust numerical way to determine rank, especially when using a tolerance for "effectively zero" values.

Concept Tested: Rank and SVD

4. In the low-rank approximation via SVD, keeping only the top \(k\) singular values minimizes:

  1. The Frobenius norm of the error
  2. The rank of the matrix
  3. The number of computations
  4. The trace of the matrix
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The correct answer is A. The truncated SVD gives the best rank-\(k\) approximation in both the Frobenius norm and the spectral norm. This is the Eckart-Young-Mirsky theorem.

Concept Tested: Low-Rank Approximation

5. QR decomposition factors a matrix as:

  1. \(A = QR\) where \(Q\) is orthogonal and \(R\) is upper triangular
  2. \(A = QR\) where \(Q\) is diagonal and \(R\) is symmetric
  3. \(A = R^TQ\)
  4. \(A = Q + R\)
Show Answer

The correct answer is A. QR decomposition expresses a matrix as \(A = QR\), where \(Q\) is an orthogonal matrix (orthonormal columns) and \(R\) is upper triangular. It is used in solving least squares and computing eigenvalues.

Concept Tested: QR Decomposition

6. The Gram-Schmidt process produces:

  1. A diagonal matrix
  2. An orthonormal basis from a set of vectors
  3. The inverse of a matrix
  4. The determinant
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The correct answer is B. The Gram-Schmidt process takes a set of linearly independent vectors and produces an orthonormal set spanning the same space. It is the foundation of QR decomposition.

Concept Tested: Gram-Schmidt Process

7. Cholesky decomposition applies to:

  1. Any square matrix
  2. Positive definite symmetric matrices
  3. Singular matrices only
  4. Rectangular matrices
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The correct answer is B. Cholesky decomposition factors a positive definite symmetric matrix as \(A = LL^T\), where \(L\) is lower triangular. It is twice as efficient as LU decomposition for applicable matrices.

Concept Tested: Cholesky Decomposition

8. The left singular vectors (columns of \(U\) in SVD) are eigenvectors of:

  1. \(A\)
  2. \(A^TA\)
  3. \(AA^T\)
  4. \(A + A^T\)
Show Answer

The correct answer is C. The left singular vectors (columns of \(U\)) are eigenvectors of \(AA^T\), while the right singular vectors (columns of \(V\)) are eigenvectors of \(A^TA\). The squared singular values are the eigenvalues.

Concept Tested: SVD and Eigenvalues

9. The pseudoinverse \(A^+\) computed via SVD satisfies:

  1. \(AA^+ = I\) always
  2. \(AA^+A = A\)
  3. \(A^+ = A^T\)
  4. \(A^+ = A^{-1}\) always
Show Answer

The correct answer is B. The Moore-Penrose pseudoinverse satisfies \(AA^+A = A\) (among other conditions). It generalizes the inverse to non-square and singular matrices, providing least-squares solutions.

Concept Tested: Pseudoinverse

10. Which decomposition is most useful for image compression?

  1. LU decomposition
  2. QR decomposition
  3. SVD with truncation
  4. Cholesky decomposition
Show Answer

The correct answer is C. Truncated SVD is ideal for image compression because it provides the optimal low-rank approximation. Keeping only the largest singular values captures the most important image features while reducing storage.

Concept Tested: SVD Applications