The Matrix Exponential

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lekh2003
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Main Question or Discussion Point

I am conducting research into the matrix exponential, and I would like to discuss the mathematical development of the exponential, and eventually some applications in terms of quantum physics.

Currently, my problem is tracking down the original usage of this technique. I am trying to find a field of mathematics where it was first used, or perhaps an article in a journal that discusses the creation of the technique from the 19th century or so. I have found many applications, such as with differential equations, but no discussion of the origins of the methodology.

Can anyone help me out?
 

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WWGD
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I believe it originated from within Lie theory. Maybe @fresh_42 knows?
 
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S.G. Janssens
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You can find a nice account of this in: Engel and Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.

Specifically, look at the contributed Chapter VII by Hahn and Perazzoli, A Brief History of the Exponential Function. (The book itself is at a graduate level, but this chapter starts before that.)
 
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lekh2003
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You can find a nice account of this in: Engel and Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.

Specifically, look at the contributed Chapter VII by Hahn and Perazzoli, A Brief History of the Exponential Function. (The book itself is at a graduate level, but this chapter starts before that.)
Thank you so much! I'll read through them.
 
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I have no idea what you mean by
I have found many applications, such as with differential equations, but no discussion of the origins of the methodology.
The exponential function is directly related to solutions of differential equations, including Lie theory. So we get back to Euler and Lagrange. It is the natural connection between tangent space (differential equations) and manifolds (solution space). Your question sounds to me as if you asked who it was who used an ansatz ##y=ae^{bx}## to solve a differential equation for the first time. My guess is Euler.
 
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StoneTemplePython
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The last couple pages of chapter 4 (i.e. section 4.7) of Stillwell's Naive Lie Theory would be of interest here.

It starts by noting that "the first person to extend the exponential function to noncommuting objects was Hamilton who applied it to quaternions almost as soon as he discovered them in 1843." (Note the matrix representation of quaternions was figured out by Cayley in 1858.)
 
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