Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACES8.40 Find the equation satisfied by the squares of the singular values of the matrixassociated with the following over-determined set of equations:2x +3y + z =0x − y − z =12x + y =02y + z = −2.Show that one of the singular values is close to zero. Determine the two largersingular values by an appropriate iteration process and the smallest one byindirect calculation.8.41 Find the SVD of⎛A = ⎝ 0 −1⎞1 1 ⎠ ,−1 0showing that the singular values are √ 3and1.8.42 Find the SVD form of the matrix⎛⎞22 28 −22⎜ 1 −2 −19⎟A = ⎝19 −2 −1⎠ .−6 12 6Use it to determine the best solution x of the equation Ax = b when (i) b =(6 − 39 15 18) T , (ii) b =(9 − 42 15 15) T , showing that (i) has an exactsolution, but that the best solution to (ii) has a residual of √ 18.8.43 Four experimental measurements of particular combinations of three physicalvariables, x, y and z, gave the following inconsistent results:13x +22y − 13z =4,10x − 8y − 10z =44,10x − 8y − 10z =47,9x − 18y − 9z =72.Find the SVD best values for x, y and z. Identify the null space of A and henceobtain the general SVD solution.8.20 Hints and answers8.1 (a) False. O N ,theN × N null matrix, ( is not ) non-singular. ( )1 0 0 0(b) False. Consider the sum ofand.0 0 0 1(c) True.(d) True.(e) False. Consider b n = a n + a n for which ∑ Nn=0 |b n| 2 =4≠ 1, or note that thereis no zero vector with unit norm.(f) True.(g) False. Consider the two series defined bya 0 = 1 , a 2 n =2(− 1 2 )n for n ≥ 1; b n = −(− 1 2 )n for n ≥ 0.Theseriesthatisthesumof{a n } and {b n } does not have alternating signsand so closure does not hold.8.3 (a) x = a, b or c; (b)x = −1; the equation is linear in x.314