GraphSlam Matrix Subtraction doubt

Question

I am trying to implement GraphSlam from this paper. I have some doubt regarding the algorithm described in this paper at table2: Algorithm GraphSLAM_linearize.

I have a doubt in line 7. As per my knowledge, Gt is a 3*3 matrix. How could I subtract 1 from a 3*3 matrix? As per matrix subtraction rule both matrices should be in same dimension.

Answer 1

That is not $[1 - G_t]$, it's $[1 \qquad -G_t]$ (exaggerating the space between the two elements. For all intents and purposes, consider $1$ to be the equivalent identity matrix, and $G_t$ is just a submatrix in this. Intuitively, in this case, $1$ would be $I_{3\times3}$.

In equation 7, the multiplication follows this order:

$(6 \times 3) * (3 \times 3) * (3 \times 6)$

Remember that the information matrix is not $(3\times3)$. It's a big, sparse matrix with non-zero submatrices along its diagonal, so adding a $(6\times6)$ matrix to that would mean you need to decompose the resulting matrix from equation 7 into submatrices that correspond to $\Omega$ at $x_t$ and $x_{t-1}$, etc.