Consider the two following observed data sets with 5 observations and 2 variables for A data...

  

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Consider the two following observed data sets with 5 observations and 2 variables for A data set and 3 variables for B A = 6 3 1 625 23 3 3 44 B = A₁ = 14.196, A₂ = 3.804, A3 = 0, 6 06 033 6 39 The eigenvalues and corresponding eigenvectors of the data B are given as follows ₁ = (-0.57735, -0.21132,-0.78868), €₂ = (0.57735, -0.78868,-0.211324) €3 = (-0.57735, -0.57735,0.57735) (a) (1 Marks) Why is not possible to get a spectral decomposition of the data set A? (b) (1 Marks) What is the possible rank of the matrix A? What is the possible rank of the matrix B? (c) (1 Marks) Check your answer by using the R function rankMatrix from the package Matrix. (d) (2 Marks) Using the above eigenvalues and eigenvectors of the ma- trix B, write down the singular value decomposition expression for B. Check the results by reconstructing B using the SVD decomposition. (e) (2 Marks) Using R provide the singular decomposition of the matrices A. Check the result by reconstructing A using the SVD decomposition. Now work with the data set A. Let A = [a₁, a2, as], ā= [₁, 2, 3] and 1 = [1, 1, 1. 1. 1] be the data matrix, sample mean column vector and all-ones column vector, respectively. 1 (f) (1 Marks) Find 7. (g) (1 Marks) Calculate 17. Show your working without using R. (h) (1 Marks) Calculate A-17. Show your working without using R. (i) (1 Marks) Calculate the SSCP (sum of squares and cross-products) matrix. Show your working without using R. (j) (1 Marks) Find S, the sample covariance matrix. (k) (1 Marks) Is S a positive definite matrix? Provide your reasoning. (1) (1 Marks) Hence using S, find R. the sample correlation matrix. Now work with the data matrix B. Let C = BTB. (m) (2 Marks) Is C a positive definite matrix? Is C a positive semidefinite matrix? Explain your answer. (n) (1.5 Marks) Let the matrix of the eigenvectors of C be U. Using R software, provide the matrix U and check that UTU = I (0) (1.5 Marks) Verify the SVD (singular value decomposition) for C using R software. (p) (1 Marks) Why the matrix C is not invertible? Consider the two following observed data sets with 5 observations and 2 variables for A data set and 3 variables for B A = 6 3 1 625 23 3 3 44 B = A₁ = 14.196, A₂ = 3.804, A3 = 0, 6 06 033 6 39 The eigenvalues and corresponding eigenvectors of the data B are given as follows ₁ = (-0.57735, -0.21132,-0.78868), €₂ = (0.57735, -0.78868,-0.211324) €3 = (-0.57735, -0.57735,0.57735) (a) (1 Marks) Why is not possible to get a spectral decomposition of the data set A? (b) (1 Marks) What is the possible rank of the matrix A? What is the possible rank of the matrix B? (c) (1 Marks) Check your answer by using the R function rankMatrix from the package Matrix. (d) (2 Marks) Using the above eigenvalues and eigenvectors of the ma- trix B, write down the singular value decomposition expression for B. Check the results by reconstructing B using the SVD decomposition. (e) (2 Marks) Using R provide the singular decomposition of the matrices A. Check the result by reconstructing A using the SVD decomposition. Now work with the data set A. Let A = [a₁, a2, as], ā= [₁, 2, 3] and 1 = [1, 1, 1. 1. 1] be the data matrix, sample mean column vector and all-ones column vector, respectively. 1 (f) (1 Marks) Find 7. (g) (1 Marks) Calculate 17. Show your working without using R. (h) (1 Marks) Calculate A-17. Show your working without using R. (i) (1 Marks) Calculate the SSCP (sum of squares and cross-products) matrix. Show your working without using R. (j) (1 Marks) Find S, the sample covariance matrix. (k) (1 Marks) Is S a positive definite matrix? Provide your reasoning. (1) (1 Marks) Hence using S, find R. the sample correlation matrix. Now work with the data matrix B. Let C = BTB. (m) (2 Marks) Is C a positive definite matrix? Is C a positive semidefinite matrix? Explain your answer. (n) (1.5 Marks) Let the matrix of the eigenvectors of C be U. Using R software, provide the matrix U and check that UTU = I (0) (1.5 Marks) Verify the SVD (singular value decomposition) for C using R software. (p) (1 Marks) Why the matrix C is not invertible?