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

Let \(V_{d-1} = \mathrm{span}(v_1, \ldots, v_{d-1})\), where \(v_1, \ldots, v_d\) 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) \(\lambda_{d-1} = \min_{0 \neq u \in V_{d-1}} R_A(u)\)
• b) \(\lambda_{d-1} = \max_{0 \neq u \in V_{d-1}} R_A(u)\)
• c) \(\lambda_{d-1} = \min_{\|u\| = 1, u \in V_{d-1}} R_A(u)\)
• d) \(\lambda_{d-1} = \max_{\|u\| = 1, u \in V_{d-1}} R_A(u)\)

Let \(W_2 = \mathrm{span}(v_{d-1}, v_d)\), where \(v_1, \ldots, v_d\) 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) \(\lambda_{d-1} = \min_{0 \neq u \in W_2} R_A(u)\)
• b) \(\lambda_{d-1} = \max_{0 \neq u \in W_2} R_A(u)\)
• c) \(\lambda_{d-1} = \min_{\|u\| = 1, u \in W_2} R_A(u)\)
• d) \(\lambda_{d-1} = \max_{\|u\| = 1, u \in W_2} R_A(u)\)

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

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

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