Suppose that T is a self-adjoint operator on the 2-dimensional complex inner product space. Suppose that the minimal polynomial of T is $$T^2-(a+c)T+(ac-|b|^2)I$$ a)Given that a, c are real numbers and b is complex. deduce from this that T is positive if and only if $$a+c\le0$$ and $$ac\le b^2$$ B) deduce from the polynomial that the eigenvalues a of T are real.
I have finished all the questions from my maths tutorial except these. We don't get answers and I am really stuck. Can somebody help me?
For part B I was thinking that eigenvalues are the roots to the minimal polynomial and using http://m.wolframalpha.com/input/?i=x%5E2-%28a%2Bc%29x%2Bac-%7Cb%7C%5E2&x=0&y=0 we can see that the roots are real so there for the eigenvalues are real. Is this a correct approach?
I have absolutely no idea how to do part a.
Thanks for any help in advance