## problem statement and notations

Following Pruessmann’s SENSE paper, the geometry factor for voxel $$i$$ in the reconstructed image is defined as

$g_i = \sqrt{[(S^H\Psi^{-1}S)^{-1}]_{ii}(S^H\Psi^{-1}S)_{ii}}$

where $$S$$ is the sensitivity matrix, and $$\Psi$$ is the noise correlation matrix.

To simplify the notation, let’s define \begin{align} A = S^H\Psi^{-1}S \end{align} From the definition, it is obvious that $$A$$ is Hermitian. Furthermore, since the noise correlation matrix $$\Psi$$ is a positive definite matrix, $$A$$ is also positive definite.

Thus the goal is to show $$A_{ii}(A^{-1})_{ii}\ge1$$ for all $$i$$, with Hermitian positive definite matrix $$A$$.

## the proof

Hermitian positive definite matrix has the spectral decomposition \begin{align} A = Q^H\Lambda Q \end{align} where $$\Lambda$$ is the diagonal matrix with positive entries, and $$Q$$ is unitary whose columns satisfy $$q_i^Hq_j=\delta_{ij}$$.

Using this decomposition and assuming the number of coils to be $$N_c$$, we have

\begin{align} g_i =& \sum_{j,k=1}^{N_c}|q_{ij}|^2|q_{ik}|^2\frac{\lambda_j}{\lambda_k} \\ =& \sum_{j=1}^{N_c}|q_{ij}|^4 +\sum_{j=1}^{N_c}\sum_{k>j} |q_{ij}|^2|q_{ik}|^2\left(\frac{\lambda_j}{\lambda_k} +\frac{\lambda_k}{\lambda_j} \right) \notag \\ \ge& \left( \sum_j |q_{ij}|^2 \right)^2 \\ = & 1 \end{align}

Here we have used the fact that all $$\lambda_j$$ are positive to get \begin{align} \frac{\lambda_j}{\lambda_k} + \frac {\lambda_k}{\lambda_j}\ge 2 \end{align}