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Ch8 算子理論與二次型
第 31 題／共 32 題
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31. Tridiagonal Matrix、Positive Definite、LDLT Decomposition
#LA-08-031
中
Tridiagonal Matrix
Positive Definite
LDLT Decomposition
📝
☆
For the 197 by 197 finite difference matrix,
A
A
A
, which statements are true?
A
197
×
197
=
[
2
−
1
0
⋯
0
−
1
2
−
1
⋱
⋮
0
−
1
2
⋱
0
⋮
⋱
⋱
⋱
−
1
0
⋯
0
−
1
2
]
A_{197 \times 197} = \begin{bmatrix} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{bmatrix}
A
197
×
197
=
2
−
1
0
⋮
0
−
1
2
−
1
⋱
⋯
0
−
1
2
⋱
0
⋯
⋱
⋱
⋱
−
1
0
⋮
0
−
1
2
A
The determinant of
A
A
A
is
D
D
D
. Then, mod(|D|,5)=3, where mod(.) is the modulo operation.
B
A
A
A
can be expressed as
A
=
B
+
C
A = B + C
A
=
B
+
C
, where
B
B
B
is a symmetric matrix and
C
C
C
is a skew-symmetric matrix.
C
A
A
A
can be expressed as
A
=
B
C
B
T
A = BCB^T
A
=
B
C
B
T
, where
B
B
B
is an orthogonal matrix and
C
C
C
is a diagonal matrix.
D
A
A
A
can be expressed as
A
=
B
C
B
T
A = BCB^T
A
=
B
C
B
T
, where
B
B
B
is a lower triangular matrix and
C
C
C
is a diagonal matrix.
E
A
A
A
's smallest eigenvalue is negative.
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