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- TL;DR Summary
- How is the transpose function of a matrix expressed?

One way to express a function of a matrix A is by a power series (a Taylor expansion). It is not too difficult to show that two functions f(A) and g(A) with such a power series representation must commute, i.e. f(A)g(A) = g(A)f(A). But matrices typically do not commute with their own transpose, so presumably the transpose function does not have convergent a power series expansion? I had not previously appreciated that even simple matrix functions may not have a power series representation. Is there another way to express the matrix transpose function, or matrix functions in general?