Operator functions and the matrix exponential
Now, we get to look at functions that involve an operator and their matrix representations. What types of functions are we talking about? Well, really any function that can be defined, such as the sine of x. You have probably never seen a function like this:
where A is a matrix. The first question is, does this even make sense? Well, mathematicians have come up with ways for this to make sense, and it has applications in quantum computing.
As we have said, if a matrix A is diagonalizable, it can be decomposed into an invertible matrix P and diagonal matrix D as:
(2)
Given that, we can represent a function involving such a matrix like so:
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