# Matrix Transpose Function

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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?

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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, ...
Yes. Transposition is a linear map, so your power series should come to an end early: ##(f(a_{ij}))_{kl} = (f_{kl}(a_{ij}))=(a_{lk})##.
... or matrix functions in general?
No. Functions in general means almost complete arbitrariness. So how should a structure on everything work? The only meaningful way is by coordinates: ##f(a_{ij})=f_{kl}(a_{11},\ldots , a_{nm})##.