Partial Solution Set, Leon
§
6.7
6.7.1
For each of the following, compute the determinants of all of the leading principal
submatrices and use them to determine whether the matrix is positive deﬁnite. (In other
words, compute the leading principal minors.)
1.
A
=
±
2

1

1
2
²
. Leading principal minors are 2 and 3; positive deﬁnite.
2.
A
=
±
3 4
4 3
²
. Leading principal minors are 3,

7; not positive deﬁnite.
3.
A
=
6 4

2
4 5
3

2 3
6
. Leading principal minors are 6, 14,

38; not positive deﬁnite.
4.
A
=
4
2
1
2
3

2
1

2
5
. Leading principal minors are 4, 8, and 13; positive deﬁnite.
6.7.5
For each of the following, ﬁnd the Cholesky decomposition
A
=
BB
T
, where
B
is a lower
triangular matrix. (Problem 6.7.4 is to ﬁnd factorizations of the form
LDL
T
, which is
an intermediate step here.)
1.
A
=
±
4
2
2 10
²
.
(d)
A
=
9 3

6
3 4
1

6 1
9
.
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 Spring '08
 Anshelvich
 Linear Algebra, Algebra, Determinant, Matrices, principal minors, positive definite

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