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Given ß = XIxnAnxnXnx1, show that the gradient of ß with respect to X has the...
In this problem, you will get more experience with taking derivatives with respect to vectors by proving common identities. In the following, it will be useful to remember that if x = (x1, . . . , xn)^⊺ and y =(y1, . . . , yn)^⊺ are vectors, then the dot product x^⊺y is a scalar equal to In this problem, you will get more experience with taking derivatives with respect to vectors by proving common identities. In the following,...
In the lectures, we introduced Gradient Descent, an optimization method to find the minimum value of a function. In this problem we try to solve a fairly simple optimization problem: min f(x) = x2 TER That is, finding the minimum value of x2 over the real line. Of course you know it is when x = 0, but this time we do it with gradient descent. Recall that to perform gradient descent, you start at an arbitrary initial point xo,...
6Show that the Eulerian strain rate is given by 147 and [see Eq. (3.5.10) for the definition of al e D. 3.5.3 Infinitesimal Rotation Tensor The displacement gradient tensor can be expressed as the sum of tensor and a skew symmetric tensor. We have of a symmetric where the symmetric part is similar to the infinitesimal strain tensor (and when l▽u| ~ |▽oul << 1), and the skew symmetric part is known as the infinitesimal rotation tensor lelilt lh the...
Question 3 In lecture, we stated that the estimate of ß in Weight Least Squares as: BWLS = (XTWX)-1xTWY Derive BW when p = 1. (It should have a form similar to simple linear regression.) Hint: Notice that we can write a weighted average as: Tw Zi=1 Hint: You may need to use weighted analogues of the sums of squares identities that we have used; you should derive (or expand) the following w2 (x - w)-w) i-1 i-1 W Question...
Equation 3.5.10 is below We were unable to transcribe this image114 KINEMATICS OF 3.5.3 Infinitesimal Rotation Tensor e displacement gradient tensor can be expressed as the sum of a tensor and a skew symmetric tensor. We have where the symmetric part is similar to the infinitesimal strain tensor (and a when VuVoul << 1), and the skew symmetric part is known as the infinitesimal rotation tensor 3.i1 We note that there is no restriction placed on the magnitude of Vu...
Please answer the following question. Please show all your working/solutions. In dealing with macroeconomic data, it is often informative to express GDP- per-capita in logs. To see the convenience, consider a variable Xt over time. The growth rate of a variable Xt from period t − 1 to period t is given by (Xt − Xt−1)/Xt−1. If we let ∆(Xt) denote the growth rate of the variable Xt from t − 1 to t, we can say that ∆(Xt) is...
1. Commutator In spherical coordinates, the gradient operator can be written as 2. 12. 12. V + ar rar sina Given the general form of the momentum operator p = , show that the commutation relations (r..] = [r.8, pel = i h, and [r, pel = 0.
Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g represents the direction in which g increases the fastest. Notice that this is the direction in the xy plane corresponding to the steepest slope up the surface, with magnitude equal to the slope in that direction. 1. At the point (2, π), find the gradient, and explain what it means. 2. Use it to construct a vector in the tangent...
show that the OLS estimator can be written as for y_i = X(beta)+epsilon_i beta hat-betat. (y (X/(xi) T))-1 УХі(epsilon).) (Hint: use the fact that (1) If a is (m, 1), and r is (m,1), t nd (2) if A is (m, m) and symmetric, and x is (m,1), then Og
6.5.8 A is a normal matrix with eigenvalues An and orthonormal eigenvectors |x,). Show that A may be written as A-Σλa) (S ). Hint. Show that both this eigenvector form of A and the original A give the same result acting on an arbitrary vector ly)