# Integrating a matrix

Is it possible to integrate a matrix?

I've been working through a problem and come up with $$\int_{t_0}^t \begin{pmatrix}\sin(s)\cdot\cos(\beta s)\\ \cos(s)\cdot\cos(\beta s)\end{pmatrix}ds$$

I'm integrating from $t_0$ to $t$

Can this be done or do we think I went wrong somewhere?

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Isn't your integrand function simply $\;\tan s\;$ ...? – DonAntonio Jul 23 '13 at 19:17
Did you mean $$\int \begin{pmatrix} \sin s & \cos (\beta s)\\ \cos s & \cos (\beta s)\end{pmatrix}\, ds$$ – Daniel Fischer Jul 23 '13 at 19:17
Ah, okay, doesn't matter much. Anyway, just integrate componentwise. – Daniel Fischer Jul 23 '13 at 19:19
It's a 2x1 matrix. I'd be greftful if you could do the edit for me! – Steve Jul 23 '13 at 19:19
After the editing I see a $\,2\times 2\;$ matrix... – DonAntonio Jul 23 '13 at 19:22

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:

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

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