Integrating a matrix

Solution 1:

Matrices form a vector space. Therefore, you can simply integrate them componentwise.

In detail. Let $A:t\mapsto A(t)$ be a function from a real interval $I$ to the space of $m\times n$ real matrices. Every entry $a_{ij}$ is a real function of a real variable. If all entries are integrable functions, then you can define the integral of the matrix as the matrix of the integrals:

$$\int A(t)\,dt := \left( \int a_{ij}(t)\,dt \right).$$

For your problem:

$$\int \begin{pmatrix}\sin(s)&\cos(\beta s)\\ \cos(s)&\cos(\beta s)\end{pmatrix}ds = \begin{pmatrix}\int\sin(s)\,ds &\int\cos(\beta s)\,ds\\ \int\cos(s)\,ds&\int\cos(\beta s)\,ds\end{pmatrix} = \begin{pmatrix}-\cos(s)&\frac{1}{\beta}\sin(\beta s)\\ \sin(s)&\frac{1}{\beta}\sin(\beta s)\end{pmatrix}. $$

There is a more sophisticated operation, in case the matrix in question belongs to a Lie algebra: ordered exponentiation. It is to integration as exponentiation is to multiplication, and permits to go from a Lie algebra element (intuitively, a differential transformation) to a group element (a whole transformation). In this case, you need a $n\times n$ matrix-valued function.

It is explained quite well here: http://en.wikipedia.org/wiki/Ordered_exponential.