Quadratic form matrix derivative confusion

Question

Assume $R$ is square-symmetrix matrix. Then we have that $$ x^T R y = y^T R x$$ Now if I take matrix derivative of these two wrt. $R$, then this would seem to imply

$$ xy^T = yx^T$$ Yet this is not true, rather they are transposes of each other.

a.) Is this because taking transpose of scalar matters when I am taking derivative wrt. a matrix

b.) If I have to take this derivative as part of larger expression, how to best decide which one to use?

score 1 · Accepted Answer · answered Apr 26 '21 at 14:39

1

Since $R$ is symmetric, $R_{ij}$ appears twice in $x\cdot Ry$ if $i\ne j$, so the correct differentiation is $x^Ty+y^Tx=y^Tx+x^Ty$, which is trivial.

answered Apr 26 '21 at 14:39

J.G.

115,835

Now, that makes sense -- thanks – Lulu Apr 26 '21 at 15:21
Didn't find these before, but for anyone interested -- more detail also in here and here and here – Lulu Apr 26 '21 at 18:21

Quadratic form matrix derivative confusion

1 Answers1