What is the QR decomposition of A?

Learning Goal: I’m working on a linear algebra multi-part question and need an explanation and answer to help me learn.1. Consider the linear system of equations A x = b where A, together with its LU decomposition, is given below: A =   −1.4916 2.3505 −0.1924 −0.7423 −0.6156 0.8886 −1.0616 0.7481 −0.7648   =   1 0 0 0.4977 1 0 0.7117 0.5180 1     −1.4916 2.3505 −0.1924 0 −1.7853 0.9844 0 0 −1.1378   and b = [1, 4, 9]>. Use the LU decomposition to determine x and show your work. (You can use Matlab/Python/a calculator to add/subtract/multiply/divide real numbers, but please don’t use a computer to solve any linear systems!)2. Consider the least squares problem: min x∈R^n ||A x − b ||_2 where A ∈ R m×n and m > n. What is the QR decomposition of A? How would one use this to solve this problem?3. (a) Write down the definition of the matrix norm ||A||_2.(b) Use the definition to show that ||AB||_2 ≤ ||A||_2||B||_2 for any A, B ∈ R n×n4. Consider the following text processing problem. Suppose that we have four documents, and we select 5 key words. The matrix X, given below, shows the number of times each keyword appears in each document. Each document corresponds to a column of X. For example, X23 = 1 means that keyword 2 appears once in document 3. X =       3 0 1 2 5 1 1 4 0 7 2 1 2 5 1 0 1 4 1 4       (a) Use the tf-idf scheme to make a new, weighted, term matrix X˜. You do not need to normalize the columns.(b) What is a non-negative matrix factorization of X˜? (You don’t have to find a non-negative factorization of X˜. Just describe what it is). How might we use this to gain insight into our data set?5. Let B be a 3 × 2 × 3 tensor with slices: B(:, :, 1) =   1 12 −4 −2 −1 0   B(:, :, 2) =   4 2 −2 2 −10 1   B(:, :, 3) =   3 0 2 31 −5 5   Write down the slices B(1, :, 🙂 and B(:, 1, :).6. Consider the graph shown in Figure 1. (a) Write down the adjacency matrix and Laplacian for this graph(b) Write pseudocode for the spectral clustering algorithm.(c) Why does spectral clustering work? Justify your answer.
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