Matrix Calculus - Identities - Identities in Differential Form

Identities in Differential Form

It is often easier to work in differential form and then convert back to normal derivatives. This only works well using the numerator layout.

Differential identities: scalar involving matrix

Condition	Expression	Result (numerator layout)

Differential identities: matrix

Condition	Expression	Result (numerator layout)
A is not a function of X
a is not a function of X
(Kronecker product)
(Hadamard product)
(conjugate transpose)

To convert to normal derivative form, first convert it to one of the following canonical forms, and then use these identities:

Conversion from differential to derivative form

Canonical differential form	Equivalent derivative form