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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1585One-Sided Diagonal Repair 5For what smallest u does the quadratic form (1+u)x 2 + 4xy + 5y 2 stay nonnegative for all (x,y)?数学困难derivation未尝试面试订阅1586Eliminate One Coordinate of a Quadratic 1Consider q(x,y) = 4x 2 + 6xy + 5y 2. If y is chosen optimally for each fixed x, what coefficient multiplies x 2 in the reduced one-variable penalty?数学简单derivation未尝试免费1587Eliminate One Coordinate of a Quadratic 2Consider q(x,y) = 7x 2 + -4xy + 3y 2. If y is chosen optimally for each fixed x, what coefficient multiplies x 2 in the reduced one-variable penalty?数学简单数值题未尝试免费1589Pick the Residual Variance in a Schur Reduction 4A coupled penalty is q(x,y) = 10x 2 + 12xy + cy 2. After optimizing out y, the desk wants the remaining x 2 coefficient to be exactly 6. What value of c is required?数学困难数值题未尝试面试订阅1590Cross-Term Budget After Eliminating One Coordinate 5For q(x,y) = 5x 2 + 2bxy + 4y 2, what is the largest absolute value |b| such that, after optimizing out y, the remaining x 2 coefficient is at least 3.75?数学困难derivation未尝试面试订阅1591Recover a Cholesky Pivot 1A positive definite matrix A = [[9, 6], [6, 13]] has Cholesky factor L = [[l11, 0], [l21, l22]] with positive diagonal. What is l22?数学简单数值题未尝试免费1592Recover a Missing Entry from a Cholesky Margin 2A matrix A = [[4, 2], [2, t]] has a Cholesky factor with positive diagonal entries 2 and 3. What is t?数学中等derivation未尝试免费1593Choose a Diagonal Entry to Hit a Cholesky Pivot 3For q(x,y) = 5x 2 + 4xy + ky 2, the Cholesky factor of the associated matrix is required to have second diagonal entry equal to 2. What value of k achieves that?数学中等derivation未尝试免费1594Residual Coefficient After Completing the Square 4Rewrite 8x 2 + 12xy + 7y 2 in the form 8(x + 3y/4) 2 + cy 2. What is c?数学困难数值题未尝试面试订阅1595Coupling Consistent With a Target Cholesky Pivot 5A matrix A = [[9, k], [k, 10]] has a Cholesky factor whose second diagonal entry is exactly 1. What values of k are possible?数学困难derivation未尝试面试订阅1596Linear Term Implied by a Target Optimizer 1For f(x) = 0.5 x T A x - b T x with A = [[4,1],[1,3]], suppose the optimizer is x* = (1,2). What vector b makes that true?数学简单derivation未尝试免费1598Second Coordinate of a Coupled Quadratic Optimizer 3For f(x) = 0.5 x T A x - b T x with A = [[6,-2],[-2,4]] and b = (8,0), what is the optimizer's second coordinate?数学中等derivation未尝试免费