1. Find the induced 2-norm, induced oo-norm, and Frobenius norm of the matrix below. A /0...
5. Let A be an n × n invertible matrix. Show that the condition number of A with respect to the Frobenius norm
5. Let A be an n × n invertible matrix. Show that the condition number of A with respect to the Frobenius norm
3. Let a >0, and for any A E Rnxn, define Aa aA (a) Prove that for any induced matrix norm, K(Ao) (b) Write the formula for det(Aa) in terms of det(A). estimating well/ill-conditioning of matrices. n(A) . Hint: examine IAall and IAal directly. (c) Based on your result from (a) and (b), comment on whether the determinant is useful for
3. Let a >0, and for any A E Rnxn, define Aa aA (a) Prove that for any induced...
(3) Define lo norm || A||00 of a matrix A. Find || A|| 10 of the following matrix A. -5 1 A= -14 0 5 0 4 Also find a vector x € R3 such that || 2 || 00 1 and A||= || Ax|0
Question B
7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i) Compute the Frobenius norm of M, i.e., where (A, B) = trace(B·A). [4 marks] 3 marks] (iii) What is NM-illp? (b) Let H be an n × n complex matrix (6) What does it mean to say that H is positive semidefinite. (il) Show that H is positive semidefinite and Hermitian if and only...
Question 2 (2 points) a mapping induced by the matrix 2] A = [1 0 lo -1 2 5 1 . Is L an isomorphism? 3] True False
1. Consider the vector space R2 with the norm || - ||p. For p = 1,2, 00, draw the unit ball B in the norm || - ||p B x R2x|p < 1} What does B, look like for some other p? Note: p has to be in the range 1 < p < oo for || - ||p to be a norm. Also, only the 2-norm is induced by an inner product on R2. (It is induced by the...
answer all steps, please
2. Let A8 28 -4 -6 13 a) Compute l|All (Frobenius norm), |Alo and ||Alli (by hand b) Compute A-1 and the norms l|All2 and IA12 in MATLAB. c Use MATLAB to compute the condition nmber in norm-2 of A, based on your results from (b). Verify your results in MATLAB using the cond command
2. Let A8 28 -4 -6 13 a) Compute l|All (Frobenius norm), |Alo and ||Alli (by hand b) Compute A-1 and...
Please follow the directions and show steps.
A2 use MATLAB, write down the MATLAB command you used. a. For the matrix A3 -4 7calculate the norm | 2 4 -1 8 0-1 b. For the matrix B0 0 4 calculate the norm | Bl22 by hand. Even the eigenvalues of BTB have to be calculated on paper without MATLAB and without using determinants!
A2 use MATLAB, write down the MATLAB command you used. a. For the matrix A3 -4 7calculate...
4. (a) Write down, without proof, all parts of the Perron-Frobenius Theorem (b) Let S be a stochastic matrix. Prove that 1 is the Perron eigenvalue of S, and e (1 Furthermore, prove that A-1 for every eigenvalue λ of S 1) is the corresponding Perron eigenvector of S (c) For each of the given matrices S(a) below determine the values of the parameter di for which the limit link oo (Si exists. Justify your answer! 1 1 2 2...
10.10 If A is an 'n x n matrix, and x is an n x 1 vector, then the linear transformation y = Ar maps* n to·m, so the linear transformation should have a condition number, condAx (x). Assume that l a subordinate norm a. Show that we can define condar (x)-[All Irl/IArll for every x 0. b. Find the condition number of the linear transformation atx [ - 2 using the oo-norm ng the oo-norm. T-3 2 1 .12...