We can still write this equation in a slightly different style. If we now relate this last result to the metric g αβ, we set B=g αβ, B -1=g αβ and det(B)=g leading to Then the determinant of this matrix, defined as the product of the elements on the main diagonal can be expressed as:
Now, if u f(x) is a function of x, then by using the chain rule, we have: d ( sin u) d x cos u d u d x. When cross-entropy is used as loss function in a multi-class classification task, then is fed with the one-hot encoded label and the probabilities generated by the softmax layer are put in. The smaller the cross-entropy, the more similar the two probability distributions are. If we now consider the exponential matrix e Aas: The derivative of sin x is cos x, The derivative of cos x is sin x (note the negative sign) and. where denotes the number of different classes and the subscript denotes -th element of the vector. Then the trace of this matrix, as for any square matrix, is the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right), so Let us consider a matrix from a general form Computes the Jacobian matrix (matrix of partial derivatives) of a given vector-valued function with respect to an appropriate list of variables.
This artilce looks at the process of deriving the variation of the metric determinant, which will be useful for deriving the Einstein equations from a variatioanl approach, in the next article Einstein-Hilbert action. WolframAlpha Widgets Overview Tour Gallery Sign In. If you like this content, you can help maintaining this website with a small tip on my tipeee page Smith and Lin (1989) used algorithmic differentiation to eva derivatives of the log-determinant of a tridiagonal matrix in linear time.
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