I'm studying multivariable-calculus and I'm trying to solve this question:
Let $GL(n,\Bbb R)$ be the group of $n×n$ invertible matrices of real numbers now let $A=\begin{pmatrix} 0 & 1 \\ 2 & 3 \end{pmatrix}$ $B=\begin{pmatrix} 1 & 2 \\ 3 & 0 \end{pmatrix}$ $C= \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ $I= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
Define $F:GL(2,\Bbb R) \to GL(2,\Bbb R)$ given by $F(X)=X^{-1}AX$
let $f:GL(2,\Bbb R) \to \Bbb R\ $given by $\ f(X)=[F(X)]_{11}\ $when $[F(X)]_{11}$ is the component in the first row and first column of the matrix $F(X)$
I need to find the second directional deriative $D_{B}D_{C}f(I)$
what I did: first I calculated $f(X)$ if $X= \begin{pmatrix}x & y \\z & w \end{pmatrix}\ $then $f(X) = \frac{-2xy+z(w-3y)}{(xw-yz)}\ $ so now I tried to find an expression for the function $D_{C}f\ $ but no matter which of the 2 definition of the directional deriative the I know I have tried both of them got me stuck because I got to a point where I needed to do alot of complex calculation and I don't think that is the point of the question. The methods I know
1.with the limit definition:$\ \lim_{t\to 0}\frac{f(X+tC)-f(X)}{t}$
2 . with the gradient : $D_{C} f = \nabla f \cdot{C}$
My question is am I right that there is an easier way to solve this question? and if so it means that my understanding of this topic is not so good, so can you refer me to some notes\videos\books that can help me get a better understanding? also if you can give me a hint on how to solve this question that will be great too
You can use the product rule on matrices to work it out.
\begin{equation} \begin{split} D_C F(X) &= (D_C(X^{-1})) AX + X^{-1}AD_C(X) \\ &= (-X^{-1}CX^{-1})AX + X^{-1}AC \\ &= -X^{-1}(F(X) + AC) \\ \end{split} \end{equation}
\begin{equation} \begin{split} D_B D_C F(X) &= -X^{-1}CX^{-1}(F(X) + AC) -X^{-1}(-X^{-1}(F(X) + AB)) \\ &= X^{-2}(F(X) + AB) - X^{-1}CX^{-1}(F(X) + AC) \end{split} \end{equation}
Evaluate at $X=I$ for the 1,1 component and you have $(I + AB - C - CAC)_{11} = 1 + 3 - 1 - 12 = -9$