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On complex derivative and complex matrices. (English) Zbl 1151.15024

The authors develop a theory of complex \(2\times 2\) matrices involving real \(2\times 2\) matrices and roots of the matrix equation \(M^2+I=0\), where \(M\) is a real \(2\times 2\) matrix and \(I\) is the multiplicative and 0 the additive identity matrix. This enables them to give an alternative definition of the complex derivative which is not a limit of a quotient and thus allows one to develop calculus in algebras that are not fields.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
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References:

[1] Anton H., Elementary Linear Algebra (1994) · Zbl 0929.15001
[2] Rudin W., Principles of Mathematical Analysis (1976) · Zbl 0346.26002
[3] Lang S., Complex Analysis (1999) · Zbl 0933.30001 · doi:10.1007/978-1-4757-3083-8
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