Let be a formal power series on having an infinite radius of convergence and fix an matrix over . The mapping carries a complex number to a complex matrix ; we can think of this as the ‘direct sum’ of different functions on , one for each entry of .

We now define the derivative of with respect to to be a mapping as follows: . In other words, thinking of as a matrix of functions, is the matrix whose entries are the derivatives of the corresponding entries of .

We will use the limit definition of derivative (that is, , where it doesnt matter how approaches 0 in ) and will assume that all derivatives exist everywhere.

Prove the following properties of derivatives:

- If , then .
- If is an matrix with constant entries (i.e. not dependent on ) then .

[My usual disclaimer about analysis applies here: as soon as I see words like ‘limit’ and ‘continuous’ I become even more confused than usual. Read the following with a healthy dose of skepticism, and please point out any errors.]

Note the following.

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= | (Now we can substitute .) | |

= | (All terms but vanish.) | |

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As desired.

Now say and ; we then have the following.

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As desired.