How do I solve this problem using eigen values?
#Linear Algebra question
elfin tulip
halcyon fog
How do I solve this problem using eigen values?
We have:
P + Q = A^4 - 5A^3 + 8A^2 - 2A = A(A^3 - 5A^2 + 8A - 2)
You can verify that the characteristic polynomial of A is p(x) = -x^3 + 5x^2 - 8x + 4. So:
P + Q = A(2 - p(A)) = 2A - A p(A)
But by Cayley-Hamilton we have p(A) = Z (zero matrix). So:
P + Q = 2A - AZ = 2A - Z = 2A
The rest is easy.
elfin tulip
Thank you so much ^_^