# computing matrix function derivatives [closed]

I was able to use the following equation to find derivatives of matrix function:

$f(X+h) = f(X) + Ah + o(|h|) \quad \cdots (1)$

where $h$ is small displacement and $A$ is the jacobian matrix. I found couple more equations that find derivatives of matrix function:

$D_Yf(X) = \lim_{t->0} \frac{f(X+tY) - f(X)}{t} \quad \cdots (2)$

$D_Yf(X) = \lim_{t->0} \frac{f(X+tY) - f(X)}{t} = tr(Y^TU) \quad \cdots (3)$

It seems like the first equation and second equation are identical with (2) being more precise in terms of the definition of a derivative( Equation (2) also shows why $|h|^2$ is ignored). Furthermore, both (1) and (2) apply to functions $\mathbb{R}^n$->$\mathbb{R}^m$. Equation (3) seems to be special case of (2). Equation (3) was used like the following:

$f(X) = tr(AX)$ $D_Yf(X) = \lim_{t->0} \frac{f(X+tY)-f(X)}{t} \\ = \lim_{t->0} \frac{tr(A(X+tY) - tr(AX)}{t} \\ = \lim_{t->0} \frac{tr(AX+AtY] - tr(AX)}{t} \\ = \lim_{t->0} \frac{tr(tAY)}{t} \\ = \lim_{t->0} tr(AY)\\ = tr(AY) \\ = tr([AY]^T) \\ = tr(Y^TA^T)$

$U=A^T$, therefore $D_Yf(X) = A^T$.

Couple questions regarding those three equations:

1. Is $h$ in eq. (1) same as $tY$ in equation (2)?(something seems a bit missing in equation(1) to me...)

2. I don't quite understand what $tr(Y^TU)$ meansin equation (3). $Y$ seems to be a directional matrix and $U$ seems to be the jacobian but what exactly does that expression mean? And how does formatting into $tr(Y^TU)$ form give us the derivative?

EDIT: they(where i found equation (3)) used column vector for the gradient. (http://www.tc.umn.edu/~nydic001/docs/unpubs/Schonemann_Trace_Derivatives_Presentation.pdf)

## closed as unclear what you're asking by whuber♦Oct 7 '17 at 16:48

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• You should read the matrix cookbook. It covers all this stuff. www2.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/… – gammer Jan 24 '17 at 5:46
• @gammer matrix cookbook seems, to me at least, like a cheat sheet with all the derivations without proof. I want to know how the cookbook derived certain stuff. In fact I was working on algebraically proving equations 33-45 in the cookbook and got stuck with this notion of deriving a trace. – MoneyBall Jan 24 '17 at 5:52
• Cool, well check it out if you want. Good luck. – gammer Jan 24 '17 at 6:20
• Because I addressed (2) in detail, starting from the definitions, in your previous question at stats.stackexchange.com/questions/257579/…, and because $tY$ in $(2)$ obviously plays exactly the same role as $h$ in $1$, I don't understand what else you need to know. – whuber Jan 24 '17 at 16:17
• @whuber okay so you're saying $tY$ is the same as $h$ so that confirms my first question. what exactly is $tr(Y^TU)$? Is U a jacobian to directional matrix Y? Then what about the trace? – MoneyBall Jan 24 '17 at 23:12