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Vector, Matrix, and Tensor Derivatives

vector , Matrix, and Tensor DerivativesErik Learned-MillerThe purpose of this document is to help you learn to take Derivatives of vectors, matrices,and higher order tensors (arrays with three dimensions or more), and to help you takederivativeswith respect tovectors, matrices, and higher order Simplify, simplify, simplifyMuch of the confusion in taking Derivatives involving arrays stems from trying to do toomany things at once. These things include taking Derivatives of multiple componentssimultaneously, taking Derivatives in the presence of summation notation, and applying thechain rule. By doing all of these things at the same time, we are more likely to make errors,at least until we have a lot of Expanding notation into explicit sums and equations for eachcomponentIn order to simplify a given calculation, it is often useful to write out the explicit formula fora single scalar elementof the output in terms of nothing butscalar variables.

vector associated with the corresponding row of the input X. Sticking to our technique of writing down an expression for a given component of the output, we have Y i;j = XD k=1 X i;kW k;j: We can see immediately from this equation that among the derivatives @Y a;b @X c;d; they are all zero unless a = c. That is, since each component of Y is ...

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