A result on the singularities of matrix functions
Author:
William T. Reid
Journal:
Quart. Appl. Math. 35 (1977), 293-296
MSC:
Primary 34B05
DOI:
https://doi.org/10.1090/qam/442340
MathSciNet review:
442340
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Abstract: For $F\left ( t \right ) = F\left ( {{t_1},...,{t_p}} \right )$ an $n \times n$ complex-valued matrix function which is continuous on an open neighborhood of ${t^0} = \left ( {{t_\alpha }^0} \right )\left ( {\alpha = 1,...,p} \right )$ and singular at ${t^0}$ there is presented a necessary and sufficient condition for $F\left ( t \right )$ to be non-singular on a deleted neighborhood of ${t^0}$. If, in addition, $F(t)$ is differentiate at ${t^0}$ then a corollary to this criterion yields a differential condition that is sufficient for such isolation of a point of singularity. Applications of corollary are given, including in particular for $n = 1$ the correction of a result stated by George and Gunderson [1].
- J. H. George and R. W. Gunderson, An existence theorem for linear boundary value problems, Quart. Appl. Math. 31 (1973/74), 127–131. MR 463554, DOI https://doi.org/10.1090/S0033-569X-1973-0463554-6
- H. F. Mathis, A theorem on the extension of a rectangular matrix of continuous functions to become a nonsingular square matrix, Proc. Amer. Math. Soc. 1 (1950), 344–345. MR 36823, DOI https://doi.org/10.1090/S0002-9939-1950-0036823-9
- William T. Reid, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0273082
- F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
J. H. George and R. W. Gunderson, An existence theorem for linear boundary value problems, Quart. Appl. Math. 31, 127–131 (1973)
H. F. Mathis, A theorem on the extension of a rectangular matrix of continuous functions to become a nonsingular square matrix, Proc. Amer. Math. Soc. 1, 344–345 (1950)
W. T. Reid, Ordinary differential equations, John Wiley and Sons, New York, 1971
F. V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, 1964
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Article copyright:
© Copyright 1977
American Mathematical Society