Understanding a matrix bound/inequality I came across the following statements;
For a positive matrix $A$, that is bounded $0 \leq A \leq I,$ where $I$ is the identity matrix and a statement like $Y \leq X$ means that $X-Y$ is positive semidefinite. The following inequality is true:
$$A(I-A)A \leq \frac{4}{27} I \tag{1}$$
The explanation as to why the inequality is true, follows from noticing that $$\max_{0\leq a \leq 1} a^2(1-a) = \frac{4}{27} \tag{2}$$
This is my question, why does (2) "explain" or justify the inequality (1). Is there a theorem or perhaps a reference in a book, one can recommend to further understand this logic? In other words I am seeking an explanation why (2) is sufficient to show (1).

If $A$ is positive semidefinite, $A$ is unitarily similar to a diagonal matrix with nonnegative diagonal entries ($0\leq A\leq I$ implying that these diagonal entries are in the interval $[0,1]$). So $A=UDU^*$, where $D$ is this diagonal matrix and $U$ is unitary. Then
$$
A(I-A)A=UDU^*(I-UDU^*)UDU^*=U[D(I-D)D]U^*.
$$
Note that $D(I-D)D$ is a diagonal matrix with the entries $d^2(1-d)$ on the diagonal, where $d$ is the corresponding diagonal entry of $D$. Now apply (2).
