Math 6010
Project 1
1. Recall that

1
A = 4
7
2
5
8

3
6 .
10
(a) rank(A) = rank(A0 A) = 3
(b) We have

A
1/2
0.4520 + 0.8030i
= 0.8993 − 0.0794i
1.6412 − 0.4571i
0.4185 + 0.1071i
1.5068 − 0.0106i
1.7779 − 0.0610i

0.7212 − 0.2798i
1.3178 + 0.0277i ,
2.5739 + 0.1593i
as can be checked by multiplying the preceding by itself. It should
not come as a surprise that the square root of A is complex: We
know that A is symmetric; therefore its eigenvalues are real. But
they need not be non negative. In fact, for our particular example,
the eigenvalues of A are 16.7075, −0.9057, and 0.1982.