# Inverse of sum of Kronecker products as a sum of Kronecker products

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## Abstract

In the context of processing global navigation satellite system (GNSS) data by least squares adjustment, one may encounter a mathematical problem when inverting a sum of two Kronecker products. As a solution of this problem, we propose to invert this sum in the form of another sum of two Kronecker products. We present and demonstrate two mathematical formulas that enable us to achieve this task. We conclude from the demonstration that there is one condition for each formula to be checked before applying this proposed matrix inversion technique. In fact, these conditions restrict greatly the application of the formulas from being more general to the inversion problems of this kind. However, when applicable, the formulas obviously save computations in general and are very useful for large and fully populated matrices. In addition, this proposed matrix inversion technique shows several benefits when used in the processing of a single baseline with multi-frequency GNSS signals. These benefits are summarized in the following. First, the fully populated variance–covariance matrix of observations is easily inverted. Second, the computation of the normal matrix becomes easier as well since the blocks in both the design and weight matrix are all written in the form of Kronecker products. Third, this proposed matrix inversion technique contributes greatly to the computation of the variance–covariance matrix of estimates.

## Keywords

Sum of Kronecker products Matrix inverse Processing GNSS data## References

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