Complexity Explorer Santa Fe Institute

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Vector and Matrix Algebra

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3.3 Differentiation as a Matrix Operation » Quiz #20 Solution

Question 1:

First, we encode the polynomial given as a vector: -x^3+4x^2-6 \Rightarrow \begin{bmatrix} -6 \\ 0 \\ 4 \\ -1 \end{bmatrix} . Differentiation is then performed by way of multiplication by the differential matrix, D= \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 &0 & 3 \\ \end{bmatrix}. Putting this together, we compute the derivative: \frac {d}{dx} [-x^3+4x^2-6] \Rightarrow \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 &0 & 3 \\ \end{bmatrix}\begin{bmatrix} -6 \\ 0 \\ 4 \\ -1 \end{bmatrix}=\begin{bmatrix} 0 \\ 8 \\ -3 \end{bmatrix} \Rightarrow -3x^2+8x.