<s>[INST] inequality-positive semidefinite matrices Let $A$ and $B$ be positive semidefinite matrices of the same size. If the largest eigenvalues of $A$ and $B$ are less than equal to $1$. probe that: $$AB+BA\geq- \frac{1}{4}I$$
In the hint, it says use the fact: $0\leq (A+B-\frac{1}{2}I)^{2}$.
In order for $ (A+B-\frac{1}{2}I)^{2}$ to be positive semidefinite, the matrix $ (A+B-\frac{1}{2}I)$ has to be positive semidefinite. The hint given above confused me because I know that $A+B$ is definitely positive semidefinite since it is the sum of two positive semidefinite matrices, but we can't be sure that $A+B-\frac{1}{2}I$ is positive semidefinite. Does anyone know how to prove that: $0\leq (A+B-\frac{1}{2}I)^{2}$?
Now assume the given hint is true, then: $0\leq A\leq I$ and $0\leq B\leq I$. It follows that:$$0\leq (A+B-\frac{1}{2}I)^{2}=A^{2}+B^{2}+\frac{1}{4}I+A(B-I)+B(A-I)\leq A+B+\frac{1}{4}I+A(B-I)+B(A-I)$$
I need to prove that:$A(B-I)+B(A-I)\leq 0$. This will be true if the product of a positive semidefinite matrix and a negative semidefinite matrix is a negative semidefinite matrix. Is this true?
 [/INST] you have $0 \le (A + B - \frac 12 I)^2$ as the left hand side is a square: 
\[ \textstyle
 \langle (A + B - \frac 12 I)^2x, x\rangle = \langle (A+B-\frac 12I)x, (A+B-\frac 12I)^*x\rangle =\langle (A+B-\frac 12I)x, (A+B-\frac 12I)x\rangle = \|(A + B -\frac 12I)x\|^2 \ge 0
\]
And for the second part we have $A^2 \le A$ as you wrote. I think you perhaps didn't remember you wanted to have $AB  + BA + \frac 14I \ge 0$, for
\begin{align*}
  \left(A + B - \frac 12\right)^2 &= A^2 + B^2 + \frac 14 I - A - B + AB + BA\\\
    &\le A + B - A - B + A B + BA + \frac 14I\\\
    &= AB + BA + \frac 14I.
\end{align*}
</s>