MMiDS 5.3: Self-Assessment Quiz

According to the variational characterization of the largest eigenvalue \(\lambda_1\) of a symmetric matrix \(A\), which of the following is true?

• a) \(\lambda_1 = \min_{\mathbf{u} \neq \mathbf{0}} R_A(\mathbf{u})\)
• b) \(\lambda_1 = \max_{\mathbf{u} \neq \mathbf{0}} R_A(\mathbf{u})\)
• c) \(\lambda_1 = \min_{\|\mathbf{u}\| = 1} R_A(\mathbf{u})\)
• d) \(\lambda_1 = \max_{\|\mathbf{u}\| = 1} R_A(\mathbf{u})\)

Let \(V_{d-1} = \mathrm{span}(\mathbf{v}_1, \ldots, \mathbf{v}_{d-1})\), where \(\mathbf{v}_1, \ldots, \mathbf{v}_d\) are the eigenvectors of a symmetric matrix \(A\) with eigenvalues \(\lambda_1 \geq \cdots \geq \lambda_d\). Which of the following characterizes the second smallest eigenvalue \(\lambda_{d-1}\)?

• a) \(\lambda_{d-1} = \min_{\mathbf{0} \neq \mathbf{u} \in V_{d-1}} R_A(\mathbf{u})\)
• b) \(\lambda_{d-1} = \max_{\mathbf{0} \neq \mathbf{u} \in V_{d-1}} R_A(u)\)
• c) \(\lambda_{d-1} = \min_{\|\mathbf{u}\| = 1, \mathbf{u} \in V_{d-1}} R_A(\mathbf{u})\)
• d) \(\lambda_{d-1} = \max_{\|\mathbf{u}\| = 1, \mathbf{u} \in V_{d-1}} R_A(\mathbf{u})\)

Let \(W_2 = \mathrm{span}(\mathbf{v}_{d-1}, \mathbf{v}_d)\), where \(\mathbf{v}_1, \ldots, \mathbf{v}_d\) are the eigenvectors of a symmetric matrix \(A\) with eigenvalues \(\lambda_1 \geq \cdots \geq \lambda_d\). Which of the following characterizes the second smallest eigenvalue \(\lambda_{d-1}\)?

• a) \(\lambda_{d-1} = \min_{\mathbf{0} \neq \mathbf{u} \in W_2} R_A(\mathbf{u})\)
• b) \(\lambda_{d-1} = \max_{\mathbf{0} \neq \mathbf{u} \in W_2} R_A(\mathbf{u})\)
• c) \(\lambda_{d-1} = \min_{\|\mathbf{u}\| = 1, \mathbf{u} \in W_2} R_A(\mathbf{u})\)
• d) \(\lambda_{d-1} = \max_{\|\mathbf{u}\| = 1, \mathbf{u} \in W_2} R_A(\mathbf{u})\)

According to the Courant-Fischer Theorem, which of the following is the local formula for the \(k\)-th eigenvalue \(\lambda_k\) of a symmetric matrix \(A\)?

• a) \(\lambda_k = \min_{\mathbf{u} \in V_k} R_A(\mathbf{u}) = \max_{\mathbf{u} \in W_{d-k+1}} R_A(\mathbf{u})\)
• b) \(\lambda_k = \max_{\mathbf{u} \in V_k} R_A(\mathbf{u}) = \min_{\mathbf{u} \in W_{d-k+1}} R_A(\mathbf{u})\)
• c) \(\lambda_k = \min_{\mathrm{dim}(V) = k} \min_{\mathbf{u} \in V} R_A(\mathbf{u}) = \max_{\mathrm{dim}(W) = d-k+1} \max_{\mathbf{u} \in W} R_A(\mathbf{u})\)
• d) \(\lambda_k = \max_{\mathrm{dim}(V) = k} \max_{\mathbf{u} \in V} R_A(\mathbf{u}) = \min_{\mathrm{dim}(W) = d-k+1} \min_{\mathbf{u} \in W} R_A(\mathbf{u})\)

According to the Courant-Fischer Theorem, which of the following is the global formula for the \(k\)-th eigenvalue \(\lambda_k\) of a symmetric matrix \(A\)?

• a) \(\lambda_k = \min_{\mathbf{u} \in V_k} R_A(\mathbf{u}) = \max_{\mathbf{u} \in W_{d-k+1}} R_A(\mathbf{u})\)
• b) \(\lambda_k = \max_{\mathbf{u} \in V_k} R_A(\mathbf{u}) = \min_{\mathbf{u} \in W_{d-k+1}} R_A(\mathbf{u})\)
• c) \(\lambda_k = \min_{\mathrm{dim}(V) = k} \min_{\mathbf{u} \in V} R_A(\mathbf{u}) = \max_{\mathrm{dim}(W) = d-k+1} \max_{\mathbf{u} \in W} R_A(\mathbf{u})\)
• d) \(\lambda_k = \max_{\mathrm{dim}(V) = k} \max_{\mathbf{u} \in V} R_A(\mathbf{u}) = \min_{\mathrm{dim}(W) = d-k+1} \min_{\mathbf{u} \in W} R_A(\mathbf{u})\)

In the Courant-Fischer theorem, what do the subspaces \(V_k\) and \(W_{d-k+1}\) represent?

• a) \(V_k\) is the span of the first \(k\) eigenvectors, and \(W_{d-k+1}\) is the span of the remaining eigenvectors.
• b) \(V_k\) is any subspace of dimension \(k\), and \(W_{d-k+1}\) is any subspace of dimension \(d-k+1\).
• c) \(V_k\) is the span of the eigenvectors corresponding to the \(k\) largest eigenvalues, and \(W_{d-k+1}\) is the span of the eigenvectors corresponding to the \(k\) smallest eigenvalues.
• d) \(V_k\) and \(W_{d-k+1}\) are arbitrary subspaces with no specific relationship to the eigenvectors.

What is the main difference between the local and global formulas in the Courant-Fischer theorem?

• a) The local formulas are easier to compute than the global formulas.
• b) The local formulas depend on a specific choice of eigenvectors, while the global formulas do not.
• c) The local formulas apply only to symmetric matrices, while the global formulas apply to any matrix.
• d) The local formulas provide upper bounds on the eigenvalues, while the global formulas provide lower bounds.

In the proof of the Courant-Fischer Theorem, which argument is used to prove the global formulas?

• a) Orthonormal basis expansion
• b) Dimension argument
• c) Variational characterization
• d) Spectral decomposition