CS513, Spring 14Prof. RonAssignment #2Due Feb. 25, 2014(1) You are given the matrixA =1 2−1 2. (ℵ)(a) Find the spectrum and the spectral radiu s of this matrix (show your work). Use then Matlab’seig and abs routines to check your answers.Note, with σ(A) the spectrum of the square matrix A, the spectral radius ρ(A) is defined byρ(A) := m ax{|λ| : λ ∈ σ(A)}.(b) Compute the 1-, 2-, and ∞- norms of A above (show your work). Use then Matlab’s normroutine to check your answers.(c) Find the left sin gu lar vectors, right singular vectors, and the singular values of A. Check that Aas well as A′map right/left singular vectors to left/right singular vectors. Based on your findingsabove, write the SVD of A, and compare it with Matlab’s command svd.(2) Let A be a symmetric m atrix.(a) For a 4×4 A of your choice, use Matlab in order to find the spectrum and eigenvectors of A andthe spectrum and eigenvectors of A′A = A2(your matrix A must be non-singular and cannot haveany zero entry). Based on that example, conjecture a general connection between the spectrumand the eigenvectors of symmetric A and the sp ectrum and eigenvectors of A′A. You will needto assume that if λ is an e-value of A, then −λ is not, so you will not cover in your conjectureevery symmetric matrix. (The conjecture should be of the form ‘(λ, v) is an eigenpair of A′A if andonly if .... is an eigenpair of A’). If you do not find any reasonable conjecture to make, run m oreexamples. However, turn in the Matlab output of one of your tests only.)(b) Prove your conjecture from (a). Note that there are two parts in the proof (the ‘if’ part andthe ‘only if’ part).(c) In view of the above, state a theorem that derives the 2-norm of a symmetric A from itsspectrum. Check your theorem against the matrix−92 144144 −8. (ℵℵ)(d) Show that your theorem in (c) does not hold, in general, for matrices that are not symmetric(for that, you simply need to provide an example).(e) What can you, thus, say about the singular values of a symmetric matrix? (Remember: singularvalues are the squareroots of the e-values of A′A.)(3) Let A be an invertible matrix.(a) Prove that (λ, v) is eigenpair of A if and only if (1/λ, v) is an eigenpair of A−1(check examplesfirst if you feel confus ed).(b) Use the claim in (a) in order to find a formu la for kA−1k2in term s of singular values of A.Explain.(4) QR factor the matrix Z on page 76 of the book. You may use Matlab to this end, butmust show all the intermediate results of the your work. (So, you cannot use the qr routine ofMatlab.)(5)(a) Describe an efficient algorithm for computing the product HA, with H a Householder matrix,and A a general matrix. You may assume that H is not given explicitly, and that, instead, theinput on H is the corresponding vector w. (Note: Multiplication of two square matrices of s izem requires O(m3) operations, despite of the fact that the number of entries is only 2m2. Youralgorithm should compute HA with O(m2) operations, w hich is best possible, since there are m2entries in A).(b) Using (a) above, write a short code that efficiently solves a square linear system of equationsusing QR factorization. Guideline: never compute any Householder matrix; simply save its vectorw. After all, the only time the Householder matrix does anything, it is being multiplied by a vectoror a matrix.(c) Run your algorithm on two examples of your choice. The two matrices you choose should haveat least order 5 each, and sh ou ld be invertible.(d) Estimate the complexity of your algorithm, i.e., the number of operations it uses as a function ofthe order of the sy stem. Note: if your algorithm is designed correctly, the complexity is determinedby the QR factorization itself, and not by the subsequent need to solve the