# Matrix exponentiation – what’s that?

Most people have heard of the exponential function that maps an arbitrary real (or even complex) number x to but what happens if x is not a number but a matrix? Does the expression with a square matrix even make sense?

The answer is: Yes, it does!

In order to understand what the expression means, we take a step back to the exponential function for scalars. When we have a look at the power series of the exponential function,

with ,

we can see that there are only multiplications, additions and divisions by a scalar involved. These operations can be generalized to matrices easily. Hence, we can define the exponential of a matrix as

.

The next question is: How can we compute the matrix exponential for a general complex matrix?

There exist several different algorithms, our project focuses on two of them: **Taylor series** and **diagonalization**.

The most intuitive one is using the representation above and replace the infinite sum by a finite one to obtain the **Taylor series**. The number of summands that one has to compute depends on the accuracy that is needed – although it is only an approximation, it serves its purpose in many applications.

The second approach to compute the exponential of a matrix in our project is **diagonalization**. At first the matrix is decomposed to a product of three matrices , and ,

,

where the columns of the matrix contain the eigenvectors of , the matrix is a diagonal matrix with the corresponding eigenvalues stored at the diagonal and is the inverse matrix of . With this decomposition, the computation of the matrix exponential is very easy because the following equality holds

.

The only expression that has not been calculated is and this matrix is again a diagonal matrix with the exponential of the diagonal entries of . If we multiply the matrices , and , we obtain the matrix exponential of .

But this is not HPC, right?

Wait and see…

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