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April 28th, 2009, 08:02 AM   #1
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Matrix Problem

Hi,

I need some help to solve the following problem.

Let A be any 2 x 2 matrix over C and let f(x) = a0 + a1x + a2.x^2 + ..... + an.x^n be any polynomial over the complex numbers C. Show that f(A) is a matrix which can be written as c1.I + c2.A for some c1, c2 belongs to C, where I is the identity matrix.

Can somebody help me how to start it?

Thanks in advance!
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April 28th, 2009, 11:13 AM   #2
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Re: Matrix Problem

If you can show it for any then you can show it for an arbitrary polynomial

Then, show that it is possible to write as , and then show by induction that it is possible for any
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April 28th, 2009, 08:57 PM   #3
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Re: Matrix Problem

the degree of minimal polynomial of is less then 2. so in polynominal ring
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April 29th, 2009, 06:49 AM   #4
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Re: Matrix Problem

Quote:
Originally Posted by mattpi
...show that it is possible to write as ...
Hint: consider the expression for an arbitrary 2x2 matrix
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