# M-<M> for M operator: why not a mismatch?

1. Mar 19, 2014

In "Quantum Computation and Quantum Information" by Nielsen & Chuang, on pp. 88-89, applying basic statistical definitions to operators, one of the intermediary steps uses the expression
M-<M>
where M is a Hermitian operator, and <M> is the expected value = <ψ|M|ψ> for a given vector ψ (that is, when one is testing for |ψ>.)
What I do not understand is how one can subtract a vector from an operator. That is, <ψ|M|ψ> is a vector, and M is an operator. For example, if one took an example of M as a 2x2 matrix and ψ as a 1x2 vector, then <ψ|M|ψ> is a 1x2 vector, and then M-<M> has a mismatch in dimensions.
What am I wrongly interpreting? Thanks.

2. Mar 19, 2014

### micromass

Staff Emeritus
Usually, if $A$ is an operator and $\lambda$ is a number, we write $A-\lambda$. This is of course nonsense, like you indicated. But what we mean by this is actually $A-\lambda I$, where $I$ is the identity operator. So numbers should often be seen as multiplied with some identity operator that we don't write.

3. Mar 19, 2014