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In-Depth Information
50
Distribution of det M(2,2,10)
45
40
35
30
25
20
15
10
5
0
−
30
−
20
−
10
0
10
20
30
det M(2,2,10)
Fig. 4.
Distribution of det
M
(2
,
2
,
10)
of the vibration becomes larger when
N
grows. Furthermore, the lower bound
of the total number can be approximately equal to a linear function, whereas
the upper bound is to a quadratic function.
Finally, the ratio of the number of matrices with zero determinant to the
total number of
M
(2
,
2
,N
) is plotted as Fig. 3. This figure also confirms the
results obtained in Sect. 5.
6.2 Statistics of Determinant
Figures 4 and 5 show the distributions of the determinant of
M
(2
,
2
,
10) and
M
(2
,
2
,
50). The distribution are symmetric, and the median and average are
exactly equal to 0. Furthermore, the number of matrices with 0 determinant
is very high, compared with other values.
Figure 6 plots the distribution of
, which suggests that the
distribution is like 1
/N
. However, it is notable that the vibration is observed
for a given determinant value.
It is also notable that since the ratio of
det
= 0 rapidly decreases as
N
grows, the number of matrices with 0 determinant becomes smaller.
Tables 3 and 4 shows the statistics of those matrices.
|
det
M
(2
,
2
,
50)
|
