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}