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This short note is devoted to the analysis of the trace of a product of two matrices in the case where one of them is the inverse of a given positive definite matrix while the other is nonnegative definite. In particular, a relation between the trace of A-1 H and the values of diagonal elements of the original matrix A is analysed.
The classical Cayley-Hamilton theorem is extended to Drazin inverse matrices and to standard inverse matrices. It is shown that knowing the characteristic polynomial of the singular matrix or nonsingular matrix, it is possible to write the analog Cayley-Hamilton equations for Drazin inverse matrix and for standard inverse matrices.
Electricity demand forecasting of an off-grid area, where no previous load data is available, is an important prerequisite for power system expansion planning. Bangladesh is a small as well as densely populated country in South Asia with a large portion of the population living under poverty line. About 48.5% of the total population has access in grid electricity. Uninterruptable power supply is one of the most important parameter for future development which ends up with a decision of obvious expansion of present grid coverage. This paper represents an analysis to forecast the electricity demand of an isolated island in Bangladesh where past history of electrical load demand is not available. The analysis is based on the identification of factors on which electrical load growth of an area depends. The forecasting has been done through inverse matrix calculation and linear regression analysis method. It has been found that the demand data, calculated from two different approaches, are close enough which spans the reliability of the proposed method. This method can be applicable for short term load forecasting of any isolated area throughout the world.
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