Geodesic
P and D in P-1XP = dg(λ, ... , λn) = D As Matrix Functions of X
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 832-837

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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). 
Rinehart, R. F. P and D in P-1XP = dg(λ, ... , λn) = D As Matrix Functions of X. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 832-837. doi: 10.4153/CJM-1966-083-0
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