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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.

Let’s again compute a scalar derivative between one component of ~y, say ~y 3 and one component of W, say W 7;8. Let’s start with the same basic setup in which we write down an equation for ~y 3 in terms of other scalar components. Now we would like an equation that expresses ~y 3 in terms of scalar values, and shows the role that W

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