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Mathematics
Linear and Abstract Algebra
Second derivative of a complex matrix
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[QUOTE="andrewkirk, post: 5916692, member: 265790"] Where did you get the following equation? I am not aware of any such identity, and when I choose an arbitrary matrix ##\phi## and calculate the two sides of the equality, they do not match: Code: [code] Phi_real <- matrix(1:4, ncol=2) / sqrt(2) Phi_Im <- matrix(8:5, ncol=2) / sqrt(2) Phi <- Phi_real + (0+1i) * Phi_Im # calculate left side of equation Conj(t(Phi)) %*% Phi # calculate right side of equation (Phi_real %*% Phi_real + Phi_Im %*% Phi_Im) / 2 [/code] Results: [code] > # calculate left side of equation > Conj(t(Phi)) %*% Phi [,1] [,2] [1,] 59+ 0i 47-18i [2,] 47+18i 43+ 0i > # calculate right side of equation > (Phi_real %*% Phi_real + Phi_Im %*% Phi_Im) / 2 [,1] [,2] [1,] 28.25 23.25 [2,] 25.25 22.25 [/code] Also, your use of Einstein Summation Notation is unfamiliar to me. As I was taught it, one only implicitly sums matching indices when one index is up and one index is down, not when both are at the same level. Further, even if one drops the 'different levels' requirement, the expression ##\delta^c_d\, \phi_{cd}^3## does not appear to validly imply a summation as it has a total of four c's and four d's. I suspect some of the later problems may relate to invalid use of implicit summation, but I cannot be sure without better understanding your intention. When in doubt, it's best to write out summation explicitly, to avoid invalid steps. [/QUOTE]
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Second derivative of a complex matrix
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