Eigenvalues Problem: Show Inverse of Diagonalizable Matrix A

[tc]

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

 

[matt grime]

By cayley-hamilton p(A)=0, now what can you do with that?

 

[mansi]

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.

 

[matt grime]

of course, we assuming that none of the eigenvalues is zero, too.

 

[tc]

o..thx matt and mansi