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3269One-PC Reconstruction of a Two-Asset MoveLet \Sigma=egin pmatrix 9&-3\-3&9\end pmatrix , whose first principal direction is along (1,-1). For the observed move x=(2,-1), what are the rank-1 reconstruction using only the first principal component and the residual?数学中等derivation未尝试面试订阅3270How Standardization Changes the PCA SpectrumA raw covariance matrix is \Sigma=egin pmatrix 9&6\6&9\end pmatrix . After standardizing each coordinate to unit variance, what are the eigenvalues of the correlation matrix, and what fraction of standardized variance does the first component explain?数学中等derivation未尝试面试订阅3272How Large Can the Third Singular Value Be?A data matrix has singular values 7, 4, and s. You plan to keep rank 2 and want to retain at least 95\% of total squared Frobenius energy. What is the largest allowable value of s?数学困难derivation未尝试面试订阅3273Which Portfolio Is Better Aligned with the Low-Variance PC?A covariance matrix has eigenvalues 25 and 1, with first eigenvector proportional to (2,1) and second eigenvector proportional to (1,-2). Compare the variances of portfolios p 1=(1,-2) and p 2=(2,1).数学中等derivation未尝试面试订阅3278How Large Must the Leading Eigenvalue Be to Reach 75%?A covariance matrix has eigenvalues \lambda 1, 3, and 2. What is the smallest value of \lambda 1 such that the first principal component captures at least 75\% of total variance?数学中等derivation未尝试面试订阅3279Recovering the Tail Singular Value from an Energy StatementA matrix has Frobenius norm 10, and its first two singular directions explain exactly 24/25 of total squared Frobenius energy. What is the smallest singular value?数学中等derivation未尝试面试订阅3281How Much of a Single-Asset Position Comes from the Market PC?Under covariance \Sigma=egin pmatrix 5&4\4&5\end pmatrix , take the portfolio p=(1,0). What fraction of its variance is attributable to the first principal component?数学中等derivation未尝试面试订阅3282Residualizing a Move Against the First PCUsing the same first principal direction u=(1,1)/\sqrt2, remove the first-PC component from the observed move r=(1,3). What residual vector remains?数学中等derivation未尝试面试订阅3283Which Singular Direction Blows Up Noise the Most?A design matrix has singular values 12, 5, and 0.5. Under the pseudoinverse, by what factor is noise amplified in the smallest-singular-value direction relative to the largest-singular-value direction?数学中等derivation未尝试面试订阅3291Exponential Decay After Two Time UnitsSolve x'(t)=-3x(t) with x(0)=12, and evaluate x(2).数学简单derivation未尝试面试订阅3293Cooling Model Over Half a UnitSolve x'(t)=-2x(t) with x(0)=10, and evaluate x( 1 2 ).数学简单derivation未尝试面试订阅3301Logistic Growth Carrying CapacityConsider the logistic ODE x'(t)=1x(t) (1- x(t) 10 ) with initial value x(0)=2. What is the long-run limit of x(t) as t ?数学简单derivation未尝试面试订阅3306Simple Harmonic Oscillator SolutionSolve y''+4y=0 with y(0)=1 and y'(0)=0.数学中等derivation未尝试面试订阅3311Stable Equilibrium of x(1-x)Consider x'(t)=x(1-x). Which equilibrium is stable?数学中等derivation未尝试面试订阅3315Stable Root in a Shifted Logistic FlowConsider x'(t)=(x-1)(3-x). Which equilibrium is stable?数学中等derivation未尝试面试订阅3316Decay of the First Sine ModeFor the heat equation u t=1u xx on 0<x< with zero boundary conditions, the initial condition is u(x,0)=\sin(1x). What is u(x,t)?数学中等derivation未尝试面试订阅3321Two-Mode Initial ConditionFor u t=1u xx on 0<x< with zero boundary conditions and initial condition u(x,0)=1\sin(1x)+2\sin(2x), write u(x,t).数学中等derivation未尝试面试订阅3323Three-Mode Heat ProfileFor u t=1u xx on 0<x< with zero boundary conditions and initial condition u(x,0)=2\sin(1x)+1\sin(2x)+1\sin(3x), write u(x,t).数学中等derivation未尝试面试订阅3325Alternating Signs in Initial HeatFor u t=1u xx on 0<x< with zero boundary conditions and initial condition u(x,0)=1\sin(1x)-2\sin(2x), write u(x,t).数学中等derivation未尝试面试订阅3331Which Fourier Mode Decays More Slowly?Under the heat equation on 0<x< , which survives longer: the mode \sin x or the mode \sin(3x)?数学中等derivation未尝试面试订阅