I have the following problem $$min \{ x^TAx + 2b^Tx + c : \|x\|_2 <=1 \}$$ where $$A \in R^{n x n}, b \in R^n, c\in R$$ I need to show that this problem can be transformed to convex optimization problem.
What I did: By Spectral decomposition theorem $$A = UDU^T$$ thus replacing it $$min \{x^TUDU^Tx + 2b^TUU^Tx + c : \|U^Tx\|_2 <=1 \}$$ where $$\|U^Tx\| = \|x\|$$. By changing variable $$y=U^Tx$$ we will get $$y^TDy + 2b^TUy + c : \|y\|_2 <=1$$
how to proceed ? I want to have problem in form of the sum of matrix elements.