Matrix derivatives cheat sheet
Kirsty McNaught
October 2017
1 Matrix/vector manipulation
You should be comfortable with these rules. They will come in handy when you want to simplify an
expression before differentiating. All bold capitals are matrices, bold lowercase are vectors.
Rule Comments
(AB)T=BTATorder is reversed, everything is transposed
(aTBc)T=cTBTaas above
aTb=bTa(the result is a scalar, and the transpose of a scalar is itself)
(A+B)C=AC +BC multiplication is distributive
(a+b)TC=aTC+bTCas above, with vectors
AB 6=BA multiplication is not commutative
2 Common vector derivatives
You should know these by heart. They are presented alongside similar-looking scalar derivatives to help
memory. This doesn’t mean matrix derivatives always look just like scalar ones. In these examples, bis
a constant scalar, and Bis a constant matrix.
Scalar derivative Vector derivative
f(x)→df
dxf(x)→df
dx
bx →bxTB→B
bx →bxTb→b
x2→2xxTx→2x
bx2→2bx xTBx →2Bx
For a more comprehensive reference, see https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.
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