- #1

Lucid Dreamer

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Hello, I am looking at a derivation that involves (note x is a column vector)

[tex] \frac {d(\vec{x}^T\vec{x})} {d\vec{x}} = \vec{x}^{T} [/tex]

So I convert to summation notation and evaluate as follows

[tex] \sum_{i,j} \frac {d(x_{i}x^{i})} {dx^{j}} [/tex]

[tex] \sum_{i,j} \frac {dx_{i}} {dx^{j}} x^{i} + \sum_{i,j} x_{i} \frac {dx^{i}} {dx^{j}} [/tex]

[tex] \sum_{i,j} \frac {dx_{i}} {dx^{j}} x^{i} + \sum_{i,j} x_{i} \delta_{ij} [/tex]

[tex] \sum_{i,j} \frac {dx_{i}} {dx^{j}} x^{i} + \sum_{i} x_{i} [/tex]

So I do recover the answer in the second term, but I am not sure what the derivative in the first term means. This derivative is of a row vector with respect to a column vector.

[tex] \frac {d(\vec{x}^T\vec{x})} {d\vec{x}} = \vec{x}^{T} [/tex]

So I convert to summation notation and evaluate as follows

[tex] \sum_{i,j} \frac {d(x_{i}x^{i})} {dx^{j}} [/tex]

[tex] \sum_{i,j} \frac {dx_{i}} {dx^{j}} x^{i} + \sum_{i,j} x_{i} \frac {dx^{i}} {dx^{j}} [/tex]

[tex] \sum_{i,j} \frac {dx_{i}} {dx^{j}} x^{i} + \sum_{i,j} x_{i} \delta_{ij} [/tex]

[tex] \sum_{i,j} \frac {dx_{i}} {dx^{j}} x^{i} + \sum_{i} x_{i} [/tex]

So I do recover the answer in the second term, but I am not sure what the derivative in the first term means. This derivative is of a row vector with respect to a column vector.

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