Abstract
The Craig–Sakamoto theorem asserts that real n × n symmetric matrices A and B satisfy \(\mbox{ det}(\mathbf{I}_{n} - a\mathbf{A} - b\mathbf{B}) = \mbox{ det}(\mathbf{I}_{n} - a\mathbf{A})\mbox{ det}(\mathbf{I}_{n} - b\mathbf{B})\) for all real numbers a and b if and only if AB = 0. In the present note a counterpart of the theorem for orthogonal projectors is established. The projectors as well as the scalars involved in the result obtained are assumed to be complex.
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Acknowledgements
The paper was prepared while the first author was visiting the Faculty of Statistics at the Dortmund University of Technology. The financial supports from the German Academic Exchange Service (DAAD) as well as from the project POKL run at the Faculty of Physics of the Adam Mickiewicz University are gratefully acknowledged. The author is also very thankful to the Faculty of Statistics for provision of excellent facilities during the stay.
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Baksalary, O.M., Trenkler, G. (2015). On a Craig–Sakamoto Theorem for Orthogonal Projectors. In: Beran, J., Feng, Y., Hebbel, H. (eds) Empirical Economic and Financial Research. Advanced Studies in Theoretical and Applied Econometrics, vol 48. Springer, Cham. https://doi.org/10.1007/978-3-319-03122-4_31
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