Consider the mapping $Q: M_n(mathbb R) rightarrow M_n(mathbb R)$ given by $Q(A) = A^2 + A – I_n$. How to determine $Q$ is smooth?

Consider the mapping $Q: M_n(mathbb R) rightarrow M_n(mathbb R)$ given by $Q(A) = A^2 + A – I_n$ (Mapping real $n$-dimensional matrices to real $n$-dimensional matrices).

How is it possible given this information to determine that $Q$ is a smooth function ?

How can I take the derivative ?

Answer

You don’t need to. You just need to remember that matrix multiplication is given by writing

$$M=begin{pmatrix} – & mathbf{v}_1 & – \
– & mathbf{v}_2 & – \
& vdots & \
– & mathbf{v}_n & -end{pmatrix},quad N=begin{pmatrix} | & | & & | \
mathbf{w}_1 & mathbf{w}_2 & ldots & mathbf{w}_n \
| & | & & |end{pmatrix}$$

Then $MN =(langle mathbf{v}_i, mathbf{w}_jrangle)$

Since the dot product is a polynomial in the entries of the matrix, it is differentiable. Similarly adding matrices results in a polynomial in all the entries. But then the whole map is a composition of differentiable maps, so the composite also is (chain rule). You take the derivative like you do for all maps between real vector spaces, you look at the partials with respect to all the coordinates, in this case since $A=(a_{ij})$ has coordinates $a_{ij}$, you’d do ${partialoverpartial a_{ij}}$.

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