I'm currently following Quantum Computation and Quantum Information by Nielsen & Chuang. I'm struggling to understand the derivation of The Heisenberg Uncertainty Principle in Box 2.4 page 89. I understand how they arrive to the following equation:
$$|\langle v|[A,B]|v\rangle|^2 \leq 4\langle v|A^2|v\rangle\langle v|B^2|v\rangle.$$
But the problem I have is with the following step. After this, they set $A = C - \langle C\rangle$ and $B = D - \langle D\rangle$ to arrive at
$$\Delta(C)\Delta(D) \geq \frac{|\langle v|[C,D]|v\rangle|}{2}.$$
What I don't understand is how you can subtract C, a matrix, with $\langle C\rangle$, a value, and set it equal to A, a Hermitian. Earlier in the book they define $\langle M \rangle = \langle v|M|v\rangle$ which is an inner product and thus a value and not a matrix. How can you subtract a value from an operator/matrix?
Edit: I now see that $\langle C \rangle$ is $\langle C \rangle 1$, but am still confused about how this substitution gets us from the first inequality to the 2nd inequality.