a***n
发帖数: 3633 | 1 对于一个nxn的矩阵A(t)，知道其关于实数t连续，且在[a,b]上处处不满秩。
那么是否存在一个nx1的连续向量x(t)，使得A(t)x(t)=0 ?
谢谢。 | s*****e
发帖数: 115 | 2 First, I would assume your question is actually about the existence of a
NONTRIVIAL x(t), otherwise, the answer is obvious.
Second, do you require that x(t) is not the zero vector for all t in [a, b]?
If so, the answer is NO.
Consider A(t) = [s(t), 0;
0, s(-t)]
where s(t)=0 for t < 0 and s(t)=t for t >= 0.
Then A(t)x(t) = 0 implies x(t)=[f(t) 0]' for t < 0 and x(t)=[0 g(t)]' for t
> 0.
Then continuity requires x(t)=[0 0]' when t = 0, i.e., f(t) --> 0 as t --> 0
- and g(t) --> 0 as t --> 0+.
【在 a***n 的大作中提到】
: 对于一个nxn的矩阵A(t)，知道其关于实数t连续，且在[a,b]上处处不满秩。
: 那么是否存在一个nx1的连续向量x(t)，使得A(t)x(t)=0 ?
: 谢谢。
| a***n
发帖数: 3633 | 3 Thanks you two. I don't need x(t) to be nonzero throughout the interval.
BTW: in what course/books such knowledge was taught?
]?
t
0
【在 s*****e 的大作中提到】
: First, I would assume your question is actually about the existence of a
: NONTRIVIAL x(t), otherwise, the answer is obvious.
: Second, do you require that x(t) is not the zero vector for all t in [a, b]?
: If so, the answer is NO.
: Consider A(t) = [s(t), 0;
: 0, s(-t)]
: where s(t)=0 for t < 0 and s(t)=t for t >= 0.
: Then A(t)x(t) = 0 implies x(t)=[f(t) 0]' for t < 0 and x(t)=[0 g(t)]' for t
: > 0.
: Then continuity requires x(t)=[0 0]' when t = 0, i.e., f(t) --> 0 as t --> 0
| s*****e
发帖数: 115 | 4 How do you know the rank of A(t) is fixed?
The rank of A(t) may not be fixed even if A(t) is C^{infinity}: just use the
same example I gave earlier and change s(t) to the C^{infinity} function:
= 0 if t <= 0;
= exp(-1/t) if t > 0.
this
t) | l********e
发帖数: 3632 | 5 线性代数里的常见问题
【在 a***n 的大作中提到】
: Thanks you two. I don't need x(t) to be nonzero throughout the interval.
: BTW: in what course/books such knowledge was taught?
:
: ]?
: t
: 0
| s*****e
发帖数: 115 | 6 The point is that even if A(t) if smooth, the rank of A(t) may change
between n-1 and 0 as t changes. The (n-1)-dimensional subspace that contains
the range of A(t) may change
abruptly even A(t) is C^{infinity} in t.
to | l********e
发帖数: 3632 | | B********e
发帖数: 10014 | 8 Kato's book on Perturbation Theory of Linear Operators
【在 a***n 的大作中提到】
: Thanks you two. I don't need x(t) to be nonzero throughout the interval.
: BTW: in what course/books such knowledge was taught?
:
: ]?
: t
: 0
|
|
