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- TL;DR Summary
- Inquiries on the applications of a function of a matrix of a single variable.

I'm glad there's a section here dedicated to differential equations.

I've seen in the fundamental theorem of linear ordinary systems, that, for a real matrix ##A##, we have ## d/dt \exp(At) = A \exp(At)##. I'm wondering if there are analogs of this, like for instance, generalizing a system of non-autonomous linear systems where the matrix ##A## is dependent on ##t##, or in other words, not a constant as in the linear systems example.

What are the applications of derivatives of function-valued matrices of a single variable, like ##A(t)##? Is there a way to generalize non-autonomous systems with this? Or otherwise, where else might derivatives of ##A(t)## occur?

I've seen in the fundamental theorem of linear ordinary systems, that, for a real matrix ##A##, we have ## d/dt \exp(At) = A \exp(At)##. I'm wondering if there are analogs of this, like for instance, generalizing a system of non-autonomous linear systems where the matrix ##A## is dependent on ##t##, or in other words, not a constant as in the linear systems example.

What are the applications of derivatives of function-valued matrices of a single variable, like ##A(t)##? Is there a way to generalize non-autonomous systems with this? Or otherwise, where else might derivatives of ##A(t)## occur?