On discrete-time Kalman filter in singular case and a kind of pseudo-inverse of a matrix†

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Abstract
The main results of this paper are as follows: (i) The filtering problem where the covariance matrices of the a priori probability distributions of the observation signals are singular is solved by neglecting unnecessary components of the observation signals. Although Kalman has already solved this problem by introducing the concept of the pseudo-inverse of a matrix, our method is simpler than that of Kalman. (ii) On the basis of the result of (i), a kind of pseudo inverse is found for any square symmetric matrix. It is shown that the use of this pseudo-inverse makes easy the calculation of the gain of the Kalman filter for the above-mentioned singular case.

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