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1553Shared-Factor Spectrum 3A = 1I + 211 T in dimension n=3. What are its eigenvalues?数学中等数值题未尝试免费1554Ridge Plus Crowd Mode 4A = 4I + 1/211 T in dimension n=6. What are its eigenvalues?数学中等derivation未尝试免费1556Mode-Weighted Matrix Power 1A diagonalizable matrix satisfies Au 1 = 5u 1 and Au 2 = 2u 2. If v = 1u 1 + -1u 2, what is A 3v?数学简单数值题未尝试免费1557Two-Mode Propagation 2A diagonalizable matrix satisfies Au 1 = 4u 1 and Au 2 = 1u 2. If v = 2u 1 + 3u 2, what is A 4v?数学中等derivation未尝试免费1558Diagonalizable Shock Propagation 3A diagonalizable matrix satisfies Au 1 = 3u 1 and Au 2 = 6u 2. If v = -1u 1 + 2u 2, what is A 2v?数学中等数值题未尝试免费1561Averaging Residual Decay 1A linear operator has dominant eigenvalue 1 and second eigenvalue 4/5. If an initial deviation from consensus is 10, what is the remaining deviation after 3 applications along that second mode?数学简单数值题未尝试免费1571Rank-One Spectrum Description 1A rank-one symmetric matrix has eigenvalue 7 along the ones direction. What are the remaining eigenvalues along all vectors orthogonal to 1?数学简单数值题未尝试免费1576Ridge Needed to Cap a Condition Number 1A symmetric positive definite matrix has eigenvalues 1 and 13. Before inverting it, a desk adds lambda I. What is the smallest nonnegative lambda that makes the condition number of A + lambda I at most 4?数学简单数值题未尝试免费1578Ridge Needed to Cap a Condition Number 3A symmetric positive definite matrix has eigenvalues 0.5 and 8. Before inverting it, a desk adds lambda I. What is the smallest nonnegative lambda that makes the condition number of A + lambda I at most 5?数学中等derivation未尝试免费1579Ridge Needed to Cap a Condition Number 4A symmetric positive definite matrix has eigenvalues 1.5 and 10.5. Before inverting it, a desk adds lambda I. What is the smallest nonnegative lambda that makes the condition number of A + lambda I at most 2.5?数学中等数值题未尝试免费1580Ridge Needed to Cap a Condition Number 5A symmetric positive definite matrix has eigenvalues 0.8 and 15.2. Before inverting it, a desk adds lambda I. What is the smallest nonnegative lambda that makes the condition number of A + lambda I at most 6?数学困难derivation未尝试面试订阅1581Worst-Case Curvature of a Coupled Penalty 1Consider q(x,y) = 3x 2 + 4xy + 8y 2. What is the largest real number m such that q(x,y) >= m(x 2 + y 2) for every pair (x,y)?数学简单数值题未尝试免费1582Worst-Case Curvature of a Coupled Penalty 2Consider q(x,y) = 5x 2 - 6xy + 5y 2. What is the largest m such that q(x,y) >= m(x 2 + y 2) for all (x,y)?数学中等derivation未尝试免费1583Cross-Term Budget Under a Curvature Floor 3For q(x,y) = 4x 2 + 2kxy + 7y 2, what is the largest absolute value |k| such that q(x,y) >= 2(x 2 + y 2) for all (x,y)?数学中等数值题未尝试免费1584Minimum Unit-Norm Penalty 4What is the minimum value of 6x 2 + 8xy + 10y 2 over all vectors satisfying x 2 + y 2 = 1?数学中等derivation未尝试免费1605Hedged Pair Variance 4Two assets have variances 1 and 9, with covariance -1. What is the variance of the portfolio with weights (1/3, 2/3)?数学困难数值题未尝试面试订阅1607Why Sample Covariance Can Be Rank DeficientWhy can a sample covariance matrix become rank deficient when the number of observations is smaller than the number of assets?数学中等essay未尝试免费1615Signed-Factor Covariance Entry 2Under a one-factor model X = bF + epsilon with factor variance 5, asset loadings (2, -1), and idiosyncratic variances (1, 4), what are Var(X 1), Var(X 2), and Cov(X 1, X 2)?数学困难derivation未尝试面试订阅1619Why Large Condition Number Destabilizes WeightsWhy does a covariance matrix with a very large condition number make optimized portfolio weights unstable?数学困难essay未尝试面试订阅1780Lasso Threshold Calibration 5A standardized lasso model has absolute scores (5.0, 4.0, 1.5). What is the smallest lambda that leaves only the strongest feature nonzero?统计困难derivation未尝试免费