[Need help] about Matrix Polynomial

buan · 2007-07-16, 15:33

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,

R.D. Silverman · 2007-07-16, 16:44

Quote:

Originally Posted by buan

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?

masser · 2007-07-16, 16:57

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...

buan · 2007-07-17, 15:07

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

2007-07-16, 15:33	#1
buan Jul 2007 2₈ Posts	[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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2007-07-16, 16:57	#3
masser Jul 2003 wear a mask 624₁₆ Posts	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...

2007-07-17, 15:07	#4
buan Jul 2007 2 Posts	Thank you for the hint Prof. it really help....