# How to get world coordinates from a 4X4 camera matrix

I have Intel T265 camera which has a camera coordinates system like following diagram:

• The camera system has two cameras and the center of the two cameras is the position of the camera system.

• The point of my interest is P, which is translated from the position of the camera system by z0 along z axis, and x0 along x axis.

What I am given is a series of 4X4 (pose) matrices coming with the camera data looking like:

    0.9988    0.0175   -0.0466         0
-0.0195    0.9989   -0.0426         0
0.0458    0.0434    0.9980         0
-0.0064   -0.0015   -0.0026    1.0000


or

   -0.1226    0.0055   -0.9924         0
-0.0438    0.9990    0.0109         0
0.9915    0.0448   -0.1223         0
-0.2279   -0.0244    0.1860    1.0000


They look like homogenous matrix (following diagram) except that they are transposed version

Now, I have a few questions about this 4x4 matrices:

1. What is the position of the camera at each time point? Are they in these 4x4 matrices?
2. Where exactly is the world coordinates and how is it oriented?
3. How do I use these 4X4 matrices to find the projected position of point P in the world coordinates system?

Or do I misunderstand anything, and how should I use these matrices? Thank you.

I have tried P_w = M'*P_c, where P_c is the center of the camera system, and M' is the transpose of the matrices I am given, but the projected positions do not look right to me when I plot it.