Author 
Message 
shna science forum beginner
Joined: 29 Mar 2006
Posts: 20

Posted: Thu Jul 13, 2006 5:34 am Post subject:
Re: The problem of derivative on scalar scailed matrix



"Peter Spellucci" <spellucci@fb04373.mathematik.tudarmstadt.de> wrote in
message news:e92gc9$kcv$1@fb04373.mathematik.tudarmstadt.de...
Quote: 
In article <e927mg$tmt$1@news.kreonet.re.kr>,
"shna" <nsh1979@postech.ac.kr> writes:
"Peter Spellucci" <spellucci@fb04373.mathematik.tudarmstadt.de> wrote in
message news:e90qj8$d9f$1@fb04373.mathematik.tudarmstadt.de...
In article <e904d7$tgj$1@news.kreonet.re.kr>,
"shna" <nsh1979@postech.ac.kr> writes:
Hi, all.
I have trouble with the following problem while reading a book about
numerical analysis.
Let H be given square matrix.
H = b M (b is scalar, M is a square matrix)
Let \lambda_i be eigenvalues of H.
Then,
d \lambda_i / d b = \lambda_i / b  (1)
(d indicates 'derivative')
Why be formula (1) correct?
Could you explain in detail?
Thank you.
From SeungHoon Na
??? homework
if M has the eigenvalues mu(j) which eigenvalues does then have b*M?
name these lam(j), hence which relation is there between lam(j) and
mu(j).
now compute (d/db) lam(j) = ??
lam(j)/b = ??
hth
peter
Thank you.
Your guideline help me solve this problem.
Here is the solution.
By applying determinant to "b M = H", we obtain
b M = H
where H = lam(1) * ... lam(d)
(d = # dimension)
After applying derivative on both sides, then
M db = H / lam(i) d lam(i)
Thus,
d lam(i) / d b = lam(i) * M / H
d lam(i) / d b = lam(i) / b
example how a false proof may lead to correct results.
if
H = b*M b scalar, H,M square matrices
then
det(H)=b^d*det(M), d=rowlength(H)

You are right. It is a mistake, but, I have never recognized).
I should have used b^d instead of b.
Quote:  but what about this:
M*x=lambda*x x not= 0
=
b*M*x = (b*lambda)*x
hth
peter

All your comments are summarized into " lambda(i) = b mu(i) " !
Then, lambda(i) has linear relation with b,
so the above formula (d lambda(i) / d b = lambda(i) / b ) is naturally
correct.
It make our problem more easy to manipulate, rather when considering the
determinant.
Thank you. 

Back to top 


Peter Spellucci science forum Guru
Joined: 29 Apr 2005
Posts: 702

Posted: Wed Jul 12, 2006 9:45 am Post subject:
Re: The problem of derivative on scalar scailed matrix



In article <e927mg$tmt$1@news.kreonet.re.kr>,
"shna" <nsh1979@postech.ac.kr> writes:
Quote: 
"Peter Spellucci" <spellucci@fb04373.mathematik.tudarmstadt.de> wrote in
message news:e90qj8$d9f$1@fb04373.mathematik.tudarmstadt.de...
In article <e904d7$tgj$1@news.kreonet.re.kr>,
"shna" <nsh1979@postech.ac.kr> writes:
Hi, all.
I have trouble with the following problem while reading a book about
numerical analysis.
Let H be given square matrix.
H = b M (b is scalar, M is a square matrix)
Let \lambda_i be eigenvalues of H.
Then,
d \lambda_i / d b = \lambda_i / b  (1)
(d indicates 'derivative')
Why be formula (1) correct?
Could you explain in detail?
Thank you.
From SeungHoon Na
??? homework
if M has the eigenvalues mu(j) which eigenvalues does then have b*M?
name these lam(j), hence which relation is there between lam(j) and mu(j).
now compute (d/db) lam(j) = ??
lam(j)/b = ??
hth
peter
Thank you.
Your guideline help me solve this problem.
Here is the solution.
By applying determinant to "b M = H", we obtain
b M = H
where H = lam(1) * ... lam(d)
(d = # dimension)
After applying derivative on both sides, then
M db = H / lam(i) d lam(i)
Thus,
d lam(i) / d b = lam(i) * M / H
d lam(i) / d b = lam(i) / b

example how a false proof may lead to correct results.
if
H = b*M b scalar, H,M square matrices
then
det(H)=b^d*det(M), d=rowlength(H)
but what about this:
M*x=lambda*x x not= 0
=>
b*M*x = (b*lambda)*x
hth
peter 

Back to top 


shna science forum beginner
Joined: 29 Mar 2006
Posts: 20

Posted: Wed Jul 12, 2006 7:17 am Post subject:
Re: The problem of derivative on scalar scailed matrix



"Peter Spellucci" <spellucci@fb04373.mathematik.tudarmstadt.de> wrote in
message news:e90qj8$d9f$1@fb04373.mathematik.tudarmstadt.de...
Quote: 
In article <e904d7$tgj$1@news.kreonet.re.kr>,
"shna" <nsh1979@postech.ac.kr> writes:
Hi, all.
I have trouble with the following problem while reading a book about
numerical analysis.
Let H be given square matrix.
H = b M (b is scalar, M is a square matrix)
Let \lambda_i be eigenvalues of H.
Then,
d \lambda_i / d b = \lambda_i / b  (1)
(d indicates 'derivative')
Why be formula (1) correct?
Could you explain in detail?
Thank you.
From SeungHoon Na
??? homework
if M has the eigenvalues mu(j) which eigenvalues does then have b*M?
name these lam(j), hence which relation is there between lam(j) and mu(j).
now compute (d/db) lam(j) = ??
lam(j)/b = ??
hth
peter

Thank you.
Your guideline help me solve this problem.
Here is the solution.
By applying determinant to "b M = H", we obtain
b M = H
where H = lam(1) * ... lam(d)
(d = # dimension)
After applying derivative on both sides, then
M db = H / lam(i) d lam(i)
Thus,
d lam(i) / d b = lam(i) * M / H
d lam(i) / d b = lam(i) / b 

Back to top 


Peter Spellucci science forum Guru
Joined: 29 Apr 2005
Posts: 702

Posted: Tue Jul 11, 2006 6:27 pm Post subject:
Re: The problem of derivative on scalar scailed matrix



In article <e904d7$tgj$1@news.kreonet.re.kr>,
"shna" <nsh1979@postech.ac.kr> writes:
Quote:  Hi, all.
I have trouble with the following problem while reading a book about
numerical analysis.
Let H be given square matrix.
H = b M (b is scalar, M is a square matrix)
Let \lambda_i be eigenvalues of H.
Then,
d \lambda_i / d b = \lambda_i / b  (1)
(d indicates 'derivative')
Why be formula (1) correct?
Could you explain in detail?
Thank you.
From SeungHoon Na

??? homework
if M has the eigenvalues mu(j) which eigenvalues does then have b*M?
name these lam(j), hence which relation is there between lam(j) and mu(j).
now compute (d/db) lam(j) = ??
lam(j)/b = ??
hth
peter 

Back to top 


shna science forum beginner
Joined: 29 Mar 2006
Posts: 20

Posted: Tue Jul 11, 2006 12:09 pm Post subject:
The problem of derivative on scalar scailed matrix



Hi, all.
I have trouble with the following problem while reading a book about
numerical analysis.
Let H be given square matrix.
H = b M (b is scalar, M is a square matrix)
Let \lambda_i be eigenvalues of H.
Then,
d \lambda_i / d b = \lambda_i / b  (1)
(d indicates 'derivative')
Why be formula (1) correct?
Could you explain in detail?
Thank you.
From SeungHoon Na 

Back to top 


Google


Back to top 



The time now is Wed Oct 18, 2017 4:43 pm  All times are GMT

