MATRICES AND VECTOR SPACES
8.40 Find the equation satisfied by the squares of the singular values of the matrix
associated with the following over-determined set of equations:
2 x+3y+z=0
x−y−z=1
2 x+y=0
2 y+z=− 2.
Show that one of the singular values is close to zero. Determine the two larger
singular values by an appropriate iteration process and the smallest one by
indirect calculation.
8.41 Find the SVD of
A=
0 − 1
11
− 10
,
showing that the singular values are√
3and1.
8.42 Find the SVD form of the matrix
A=
22 28 − 22
1 − 2 − 19
19 − 2 − 1
−612 6
.
Use it to determine the best solutionxof the equationAx=bwhen (i)b=
(6 −39 15 18)T, (ii)b=(9 −42 15 15)T, showing that (i) has an exact
solution, but that the best solution to (ii) has a residual of√
18.
8.43 Four experimental measurements of particular combinations of three physical
variables,x,yandz, gave the following inconsistent results:
13 x+22y− 13 z=4,
10 x− 8 y− 10 z=44,
10 x− 8 y− 10 z=47,
9 x− 18 y− 9 z=72.
Find the SVD best values forx,yandz. Identify the null space ofAand hence
obtain the general SVD solution.
8.20 Hints and answers8.1 (a) False.ON,theN×Nnull matrix, isnotnon-singular.
(b) False. Consider the sum of(
10
00
)
and(
00
01
)
.
(c) True.
(d) True.
(e) False. Considerbn=an+anfor which∑N
n=0|bn|(^2) =4= 1, or note that there
is no zero vector with unit norm.
(f) True.
(g) False. Consider the two series defined by
a 0 =^12 ,an=2(−^12 )n for n≥1; bn=−(−^12 )n for n≥ 0.
Theseriesthatisthesumof{an}and{bn}does not have alternating signs
and so closure does not hold.
8.3 (a)x=a,borc;(b)x=−1; the equation is linear inx.