Matrix Integral

The integral of a matrix is found by integrating each element of the matrix individually.

The integral of a matrix $ A(t) $, with dimensions $ m \times n $ depending on a variable $ t $, is:

$$
\int A(t) \, dt = \begin{pmatrix}
\int a_{11}(t) \, dt & \dots & \int a_{1n}(t) \, dt \\
\int a_{21}(t) \, dt & \dots & \int a_{2n}(t) \, dt \\
\vdots & \ddots & \vdots \\
\int a_{m1}(t) \, dt & \dots & \int a_{mn}(t) \, dt
\end{pmatrix} $$

Each element $ a_{ij}(t) $ of the matrix is integrated as if it were a scalar function.

As a result, the integral is a matrix of the same dimensions $ m \times n $ as $ A(t) $, with each element representing the integral of the corresponding original element.

A Practical Example

Let’s consider a matrix $ A(t) $ dependent on $ t $:

$$ A(t) = \begin{pmatrix} 2t & t^2 \\ \sin(t) & e^t \end{pmatrix} $$

This is a simple 2x2 square matrix with two rows and two columns.

We compute the integral of each element with respect to $ t $:

$ \int a_{11} \, dt = \int 2t \, dt = t^2 + C_1 $
$ \int a_{12} \, dt = \int t^2 \, dt = \frac{t^3}{3} + C_2 $
$ \int a_{21} \, dt = \int \sin(t) \, dt = -\cos(t) + C_3 $
$ \int a_{22} \, dt = \int e^t \, dt = e^t + C_4 $

Here, $ C_1, C_2, C_3, C_4 $ are the constants of integration for each integral and should be treated as distinct values.

Thus, the integral of the matrix $ A(t) $ is:

$$ \int A(t) \, dt = \begin{pmatrix} t^2 + C_1 & \frac{t^3}{3} + C_2 \\ -\cos(t) + C_3 & e^t + C_4 \end{pmatrix} $$

And so forth.