Digital Signal Processing Reference
In-Depth Information
(a) What are the eigenvalues
λ
1
,λ
0
of
R
2
?
(b) Express the eigenvalues
η
2
,η
1
,η
0
of
R
3
in terms of
a
.
(c) Show that
a
2
1
2
Tr (
R
−
1
)=
2
1
3
Tr (
R
−
1
11
−
3
,
)=
2
3
6(3
−
a
2
)
You can use the fact that the trace is the sum of eigenvalues.
(d) Show that
R
2
is positive definite, and that
R
3
is positive definite if
and only if
a
2
<
3
.
(e) For 0
<a
2
<
1
/
3showthat
1
3
Tr (
R
−
1
)
<
1
2
Tr (
R
−
1
)
,
3
2
though
R
3
and
R
2
are both positive definite.
Thus, even though the positive definite matrix
R
3
has
R
2
as its leading
principal submatrix, the inequality (8.28) is violated.
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