s**********2
发帖数: 61 | 1 Given a matrix A=(1.5,0.5,0)
(0.5,1.5,0)
(0, 0, 3)
Find a matrix S & a diagonal matrix D such that A=SDS^-1, where S^-1 is the
inverse of S. Also, choose S such that S^-1 = S' (S' is the transpose of S)
I know that D is a similar matrix to A, & have found it, but I am having
trouble finding S. I'm trying to find the basis w/ eigenvectors, but I find
that route gives me an uninvertible matrix. Any suggestions as to what I am
doing wrong? | b*********n
发帖数: 56 | 2 That's impossible. A has three distinct eigenvalues, and each e-val has one
eigenvector. So these three eigenvectors forms S. This S must be invertible.
Here is the result from matlab:
S =
-0.7071 0.7071 0
0.7071 0.7071 0
0 0 1.0000
D =
1 0 0
0 2 0
0 0 3 | s**********2
发帖数: 61 | 3 I got the same D value, and found out what I was doing wrong w/ the
eigenvectors, but I still am not getting the S value you did from Matlab. I
get the following e-vectors:
x=(1,-1,0)
y=(1,1,0)
z=(1,3,1)
These don't form the right basis for S. Any further suggestions? Thanks
one
invertible.
【在 b*********n 的大作中提到】
: That's impossible. A has three distinct eigenvalues, and each e-val has one
: eigenvector. So these three eigenvectors forms S. This S must be invertible.
: Here is the result from matlab:
: S =
: -0.7071 0.7071 0
: 0.7071 0.7071 0
: 0 0 1.0000
: D =
: 1 0 0
: 0 2 0
| b*********n
发帖数: 56 | 4 "where S^-1 is the inverse of S. Also, choose S such that S^-1 = S' (S' is
the transpose of S)"
This requires that S be orthonormal.
About what you get:
Firstly, z seems wrong, because it is not orthogonal to x or y;
Secondly, after you get x, y, z right, you need to normalize them, that is,
to make each of them be of norm 1. | s**********2
发帖数: 61 | 5 我明白了，谢谢！
,
【在 b*********n 的大作中提到】
: "where S^-1 is the inverse of S. Also, choose S such that S^-1 = S' (S' is
: the transpose of S)"
: This requires that S be orthonormal.
: About what you get:
: Firstly, z seems wrong, because it is not orthogonal to x or y;
: Secondly, after you get x, y, z right, you need to normalize them, that is,
: to make each of them be of norm 1.
|
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