Derivative of matrix trace

Asked Jun 12 '20 at 12:30

Active Jun 12 '20 at 12:59

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I have konwn the derivative of matrix trace to matirx. However, is there a expression representing the derivative of matrix trace to vector.

For instant, matrix $P\in \mathbb{R}^{d\times n}$ and vector $x\in \mathbb{R}^{d}$. $d,n \in \mathbb{R}^{+} $.

$f(x)=trace(PP^Txx^T)$.

Question: What is the answer of $\frac{\partial f(x)}{\partial x}$?

Thanks a lot! : )

edited Jun 12 '20 at 12:59

asked Jun 12 '20 at 12:30

heng zhu

where do you get stuck? – user251257 Jun 12 '20 at 12:33
4

$\newcommand{\tr}{\operatorname{trace}}$Note: $x^T PP^T x$ is just a $1\times1$ matrix (can be treated as a scalar), so $\tr\left(x^T PP^T x\right)=x^T PP^T x$. Then see e.g. How to take the gradient of the quadratic form?. – Minus One-Twelfth Jun 12 '20 at 12:34

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