Suppose that the matrix X contains the exponential windowing data and that a new data vector x arrives. Give a formula for the new exponential windowing data matrix XX* and show that the cost of computing it from X and x is 0(m²) multiplications.
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- Let f ∈ C+ 2π with a zero of order 2p at z. Let r>p and m = n/r. Then there exists a constant c > 0 independent of n such that for all nsufficiently large, all eigenvalues of the preconditioned matrix C−1 n (Km,2r ∗ f)Tn(f) are larger than c.2. Show the following cordinate frame matrix for cordinate space that has the following transform applied to it [order given, in global/world space]: rotation around the X axis of 30 degrees translation along (X=20,Y=10,Z=50) rotation around the Y axis of 135 degrees scaling factor of 6 along the Z axis translation along (X=50,Y=0,Z=50)a. Build an adjacency matrix ? for this map. b. How many paths of length 2 from V5 to V1 exist? c. How many paths of length 3 from V5 to V1 exist?
- Bu = ƒ and Cu = f might be solvable even though B and C are singular. Show that every vector f = Bu has ƒ1 + ƒ2+ ……. +fn = 0. Physical meaning: the external forces balance. Linear algebra meaning: Bu = ƒ is solvable when ƒ is perpendicular to the all – ones column vector e = (1, 1, 1, 1…) = ones (n, 1).we represent the finite-length signals as vectors in Euclidean space, many operations on signals can be encoded as a matrix-vector multiplication. Consider for example a circular shift in C: a delay by one (i.e. a right shift) transforms the signal x = (xo X1 X2]" into x = [xı xo xz]" and it can be described by the matrix TO D = [0 1 0'0 0 1'1 0 0] so that x = Dx. Determine the matrix F that implements the one step difference operator in C ie the operator that transforms a signal x into [(x - x)(x1 - x0)(x2 - 1)]Using Matlab, A=[1 2 3;4 5 6;7 8 10]; a)Create a vector version of the original matrix A. b)Extract those entries of the original matrix A that are greater than 5
- Suppose a 2D bounding box is given by (≤ is "less than or equal to"): -1 ≤ x ≤ 1-10 ≤ y ≤ 10 Compute the new Bounding Box for the transformation given by the following matrix: (see section 9.4.4) 0.866 -0.5 0.5 0.866 Now enter the xmax for the new Bounding Box (round it to 3 decimal places, if needed).(20pts) Consider the matrixA =−2 11−10 5(a) Determine, by hand, an SVD of A, A = UΣVT. The SVD is not unique,so find the one with the minimum number of minus signs in U and V . List thesingular values σ1 and σ2, and the left and right singular vectors.(b) Find A−1 not directly, but via the SVD. Check your result by typing inv(A)in MATLAB or calculating A−1 by other methods.(c) Find the eigenvalues λ1 and λ2 of A. Verify that detA = λ1λ2 and |detA| =σ1σ2Consider the following 10 data points: X= {(2,0, 1, -3, -2), (0, 2, -3, -3,-2), (1,2,1,3,-2), (-1, 1, 3, 2,-1), (1, 0, 1,-1, 1), (2, 3,-1,1,-2), (-2,3,-3, 3, 2), (-2,-2,2,3,-2), (-2,-3, 1, -2, -3). (-3, 2, 0,-1,-2)}. Compute the unit length principle components of X and choose two of them for PCA, then calculate the projection of each data on these two principal components. You could use python or matlab to obtain eigenvectors and eigenvalues.
- Consider a vector ('L', 'M', 'M', 'L', 'H', 'M', 'H', 'H'), where H stands for ‘high’, L stands for ‘low’, and M for ‘medium’ in rstudio. a. Convert this vector into an unordered factor. b. Convert this vector into an ordered factor with Low < Medium < High.Write Algorithm for Unsupervised Feature Selection with 2,1–2Input: data matrix X, hyper-parameters α and β, penalty parameter ρUse the outer product version of Hebb rule learning to find the weight matrix inbipolar form for the bidirectional associative memory network based on the followingbinary input-output vector pairs:s(1) = (1 0 0 0) 41) = (1 0)s(2) = (1 0 0 1) t(2) = (1 0)s(3) = (0 1 0 0) t(3) = (0 1)s(4) = (0 1 1 0) t(4) = (0 1)b. Using the unit step function (with threshold 0) as the output units' activationfunction, test the response of your network on each of the input patterns. Describethe results obtained.c. Test the response of your network on various combination of input patterns with"mistakes" or "missing data" (as in Example 3.24). Discuss the results you observe.