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P and D in P-1XP = dg(λ, … , λn) = D As Matrix Functions of X

Published online by Cambridge University Press:  20 November 2018

R. F. Rinehart
R. F. Rinehart*
Affiliation:
U.S. Naval Postgraduate School, Monterey, California
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Let be the algebra of n X n matrices over the complex field C, X = ‖xrs‖ a matrix of , and f(X) = ‖(frs(x11, x12, … , xnn)‖ a function with domain and range in . If the frs are differentiable with respect to each of the xij on some open set R of , then the differential df(X) = ‖dfrs(xij)‖ exists, and, moreover, f(X) is Hausdorff-differentiable (HD) (1, 3, 7) i.e. df(X)is expressible in the form

where dX = ‖dxrs‖, and the matrices Ai Bi are independent of dX. The Hausdorff derivative fI(X) is defined to be

i.e. the value of df(X) for dX = I, the identity matrix (2).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 832 - 837
Copyright
Copyright © Canadian Mathematical Society 1966

References

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