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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3280What Does Whitening Mean When the Covariance Is Singular?Suppose \Sigma=egin pmatrix 2&2\2&2\end pmatrix . Explain what can and cannot be whitened, and identify the subspace where a pseudo-whitening transform still makes sense.数学中等essay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3284Why Centering Matters Before PCAWhy can skipping mean-centering make the first principal component mostly track the mean level instead of genuine variation?数学中等essay未尝试面试订阅3285Why Whitening Can Hurt Noisy Small-Eigenvalue DirectionsWhy can whitening make downstream models more fragile when some covariance eigenvalues are tiny?数学中等essay未尝试面试订阅3286Why Sign Flips of PCs Do Not MatterWhy are principal components only identified up to sign?数学中等essay未尝试面试订阅3287Why PCA Can Miss a Predictive DirectionWhy can a low-variance direction be more useful for prediction than the first principal component?数学简单essay未尝试面试订阅3288Why Whitening Can Hurt Economic InterpretabilityWhy can whitening be numerically useful while simultaneously making a factor model harder to interpret economically?数学中等essay未尝试面试订阅3289Why Truncated SVD Is a Structured DenoiserWhy does low-rank truncation often denoise a matrix instead of merely compressing it?数学中等essay未尝试面试订阅3290Why Similar Spectra Can Hide Different SubspacesWhy do similar explained-variance ratios not guarantee that two datasets have learned the same principal directions?数学中等essay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3296Linear ODE with Constant ForcingSolve x'(t)+2x(t)=6 with x(0)=1, and simplify the solution.数学中等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