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Eigenvalues problem

  1. Nov 21, 2004 #1

    tc

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    let A be a diagonalizable matrix with eignvalues = x1, x2, ..., xn
    the characteristic polynomial of A is
    p (x) = a1 (x)^n + a2 (x)^n-1 + ...+an+1
    show that inverse A = q (A) for some polynomial q of degree less than n
     
  2. jcsd
  3. Nov 21, 2004 #2

    matt grime

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    By cayley-hamilton p(A)=0, now what can you do with that?
     
  4. Nov 21, 2004 #3
    Yeah, you use cayley-hamilton theorem
    so, you have p(A)=0...
    That implies a*A^n + b*A^(n-1) + c*A^(n-2)........+ I= 0
    (i've used a,b,c as coefficients)...then take the identity matrix to the other side. Multiply both sides by inverse of A. Then RHS becomes -A^(-1) the LHS shows that the characteristic polynomial is of degree < n.
     
  5. Nov 21, 2004 #4

    matt grime

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    of course, we assuming that none of the eigenvalues is zero, too.
     
  6. Nov 22, 2004 #5

    tc

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    o..thx matt and mansi
     
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