Re: eigenvector for a 3-by-3 symmetric real matrix
- From: "Alex. Lupas" <alex.lupas@xxxxxxxxx>
- Date: 5 Apr 2007 03:17:33 -0700
On Apr 5, 11:05 am, "Jeremy Watts" <jwatts1...@xxxxxxxxxxx> wrote:
"agou" <agou....@xxxxxxxxx> wrote in message=================
news:1175746181.292952.303120@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
hi, there
i wanna calculate the eigenvector to the smallest eigenvalue for a 3-
by-3 symmetric positive definite real matrix. i know it might sound
silly. however, i wanna put this stuff running on hardware. therefore,
analytic result would be best. however, it's not trivial. therefore,
maybe an iterative but fast converging algorithm will be preferred. QR
looks like too complicate either. does anyone have any better idea or
book i could refer to?
thanks a lot! regards zhaoyi
Assume that
A=
[a_{1,1} a_{1,2} a_{1,3}]
[a_{2,1} a_{2,2} a_{2,3}]
[a_{3,1} a_{3,2} a_{3,3}]
is an arbitrary 3 x 3 real matrix.
Then the eigenvalues r_1,r_2,r_3 are the roots
of charcteristic equation
det(A-r*I)=0 which is equivalent to
P(r):=r^3-s*r^2+u*r -d =0
where
s=tr(A):= a_{1,1}+a_{2,2}+a_{3,3}
u: =sum of three 2x2 determinants=
|a_{2,2} a_{2,3}|
|a_{2,2} a_{2,3}|+
|a_{1,1} a_{1,3}|
|a_{3,1} a_{3,3}|+
|a_{1,1} a_{1,2}|
|a_{2,1} a_{2,1}|
and d:=det(A).
Further assume that r_1,r_2,r_3 are real,
e.g. in case when A is symmetric. More
precisely consider
r_3 =<r_2 =<r_1 .
Find D:=2*s^2-6*u . Note that D>= 0 .
Denote x:= (r_1+r_2+r_3)/3 = s/3 .
Then
sqrt(D/2) =< |r_3-r_1|=< sqrt(2*D/3)
Q(1):=x + sqrt(D*0.5)/3 =< r_1 =< S(1):=x+sqrt(2*D)/3
Q(2):=x-sqrt(D*0.5)/3 =< r_2 =< Q(1)
Q(3):=x-sqrt(2*D)/3 =< r_3 =<Q(2)
In order to find the eigenvalues apply an iterative method,
like Newwton, that is
x_{n+1}=x_n - f(x_n)/f'(x_n) , n=0,1,..., x_0:=r_0 .
Select as starting points r_0 one of numbers
{S(1),Q(1),Q(2),Q(3) }.
Let us note that, according to Fourier,
r_0 is "good selected " when f(r_0)*f"(r_0) > 0 .
.
- References:
- eigenvector for a 3-by-3 symmetric real matrix
- From: agou
- Re: eigenvector for a 3-by-3 symmetric real matrix
- From: Jeremy Watts
- eigenvector for a 3-by-3 symmetric real matrix
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