# Matrix Exponential: Researching Origins & Applications

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• lekh2003
In summary, the matrix exponential has its origins in Lie theory and can be traced back to Euler's use of the ansatz y=ae^{bx} to solve differential equations. It was later extended to noncommuting objects by Hamilton in 1843 and applied to quaternions. A good source for more information on the history of the exponential function is the contributed Chapter VII by Hahn and Perazzoli in Engel and Nagel's book "One-Parameter Semigroups for Linear Evolution Equations."
lekh2003
Gold Member
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?

I believe it originated from within Lie theory. Maybe @fresh_42 knows?

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.)

lekh2003
S.G. Janssens said:
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.

S.G. Janssens
I have no idea what you mean by
lekh2003 said:
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.

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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lekh2003

## 1. What is the Matrix Exponential?

The Matrix Exponential is a mathematical concept that involves raising a square matrix to a power. It is similar to the exponential function in calculus, but instead of a scalar value, it operates on a matrix. The result of the Matrix Exponential is also a matrix.

## 2. What are the origins of Matrix Exponential?

The Matrix Exponential was first introduced by the mathematician John von Neumann in the 1930s. It was further developed by other mathematicians such as Olga Taussky-Todd and Gene Golub in the 1950s and 1960s. It has since been used in various fields such as physics, engineering, and computer science.

## 3. What are the applications of Matrix Exponential?

The Matrix Exponential has a wide range of applications in different fields. It is commonly used in solving systems of linear equations, analyzing dynamical systems, and calculating probability distributions in Markov chains. It is also used in computer graphics, signal processing, and quantum mechanics.

## 4. How is Matrix Exponential researched?

Matrix Exponential is researched through mathematical analysis and computer simulations. Mathematicians and scientists use various techniques such as eigenvalue decomposition, Taylor series expansion, and numerical methods to study the properties and behaviors of Matrix Exponential. They also conduct experiments to validate their findings.

## 5. What are the future developments in Matrix Exponential research?

There are ongoing research efforts to further understand and improve the applications of Matrix Exponential. Some potential future developments include finding more efficient algorithms for computing the Matrix Exponential, exploring its applications in machine learning and data analysis, and investigating its connections to other mathematical concepts such as graph theory and differential equations.

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