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

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In summary, the conversation discusses the expression M-<M> where M is a Hermitian operator and <M> is the expected value for a given vector ψ. The confusion arises from subtracting a vector from an operator, but it is clarified that the vector should be seen as multiplied with an identity operator.
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nomadreid
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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.
 
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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.
 
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Thanks, micromass. That clears it up completely.
 

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

What is the purpose of using the "M for M" operator?

The "M for M" operator, also known as the mismatch operator, is used in scientific research to quantify the degree of difference between two sets of data. It is often used in genetics and molecular biology to compare DNA sequences and identify genetic variations.

How does the "M for M" operator work?

The "M for M" operator works by comparing two sets of data, usually DNA sequences, and identifying the number of differences between them. It assigns a numerical value to represent the degree of mismatch between the two sets of data.

Why is the "M for M" operator preferred over other methods of comparing data?

The "M for M" operator is preferred over other methods of comparing data because it is a more accurate and sensitive measure of differences. It takes into account all mismatches, including insertions, deletions, and substitutions, whereas other methods may only consider one or two types of differences.

Can the "M for M" operator be used for data other than DNA sequences?

Yes, the "M for M" operator can be used for any type of data that can be compared and quantified. It is commonly used in fields such as linguistics, economics, and computer science to measure the degree of mismatch between data sets.

Are there any limitations to using the "M for M" operator?

Like any scientific method, the "M for M" operator has its limitations. It may not be suitable for highly complex data sets or when comparing data from different sources. It is important to carefully consider the data and the research question before using the "M for M" operator.

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