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Old 2007-07-16, 15:33   #1
buan
 
Jul 2007

102 Posts
Default [Need help] about Matrix Polynomial

Can any one give a hint...
I want to prove that for Matrix A if all of its eigenvalue is less than one in magnitude ( |eigenvalue| < 1 ), then:

I + A + A^2 + A^3 + ... = inverse of matrix (I-A)

thanks,
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Old 2007-07-16, 16:44   #2
R.D. Silverman
 
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Nov 2003

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Quote:
Originally Posted by buan View Post
Can any one give a hint...
I want to prove that for Matrix A if all of its eigenvalue is less than one in magnitude ( |eigenvalue| < 1 ), then:

I + A + A^2 + A^3 + ... = inverse of matrix (I-A)

thanks,
Hint:

Simply expand I/(I-A) and ask what are the requirements for convergence?
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Old 2007-07-16, 16:57   #3
masser
 
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Another hint: What is (I-A)*(I+A+A^2+A^3+....) ?

To reiterate what Dr. Silverman says above, you do need to show that the series converges...
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Old 2007-07-17, 15:07   #4
buan
 
Jul 2007

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Thank you for the hint Prof. it really help....
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