Q4 Suppose we want to find a vector a to project X E RP to 1-dimensional...
Consider a random vector X e RP with mean EX is a p x p dimensional matrix. Denote the jth eigenvalue and jth eigenvector of as and øj, respectively. 0 and variance-covariance matrix Cov[X] = . Note that Define the random score vector Z as Х,Ф — Z where is the rotation matrix with its columns being the eigenvectors 0j, i.e., | 2|| Ф- Perform the following task: Show that the variance-covariance matrix of random score vector Z is ....
Q4. Let 1.01 0.99 0.99 0.98 (a) Find the eigenvalue decomposition of A. Recall that λ is an eigenvalue of A if for some u1],u2 (entries of the corresponding eigenvector) we have (1.01 u0.99u20 99u [1] + (0.98-A)u[2] = 0. Another way of saying this is that we want the values of λ such that A-λ| (where I is the 2 x 2 identity matrix) has a non-trivial null space there is a nonzero vector u such that (A-AI)u =...
4. Let x- be a two dimensional feature vector (a) Suppose that we collect the following four measurements for an input belonging to class a x1 -6 -10 10 -6 Assuming that the class conditional density is Gaussian, find the maximum likelihood estimates of the mean vector and covariance matrix for class ω (b) Suppose that data come from two classes, a, and co,. Assume that . The a priori class probabilities are equal . The class conditional densities are...
Suppose that T E End(V) where V is some 3-dimensional real vector space. Suppose that det(T) 1 and that i is an eigenvalue of T. Find a, b, and C such that T5 = aT2 + 6T +c.
Let A be an invertiblen x n matrix and be an eigenvalue of A. Then we know the following facts. 1) We have jk is an eigenvalue of A* 2) We have 1 -1 is an eigenvalue of A-1 If 1 = 5 is an eigenvalue of the matrix A, find an eigenvalue of the matrix (A? +41) -'. Enter your answer using three decimal places. Hint: First find an eigenvalue of A² +41. You might do this by assuming...
Problem 2: Given a collection of data { zNJS R" we define 1. The sample mean of the points is given by 2. The sample variance of the points is given by N 2 3. The covariance matrix of the points is given by Suppose that (N) S R is a collection of data points. Using Lagrange Multipliers, show that the unit vector w for which the set (i.N), where wy, has maximum variance is the normalized eigenvector of Cov(ia)...
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
7. [2p] (a) In a two-dimensional linear space X vectors el, e2 formi a basis. In this basis a vector r E X has expansion x = 2e1 + e2. Find expansion of the vector x in another basis 1 -2 er, e2, of X, if the change of basis matrix from the basis e to the basis e, s (b) In a two-dimensional linear space X vectors el, e2 forn a basis. In this basis a vector r E...
1.4 Suppose x is a random vector drawn from a d-dimensional multivariate Gassian distribution with mean 0 and covariance Σ Define y := Qx + u, for a known (invertible) d × d matrix Q, and a dx 1 vector v. What is the distribution of y?
Suppose X is a random vector, where X = (X(1), . . . , x(d))T , d with mean 0 and covariance matrix vv1 , for some vector v ER 1point possible (graded) Let v = . (i.e., v is the normalized version of v). What is the variance of v X? (If applicable, enter trans(v) for the transpose v of v, and normv) for the norm |vll of a vector v.) Var (V STANDARD NOTATION SubmitYou have used 0...