# Is the Kronecker sum an affine transformation?

Is the Kronecker sum

$$A \oplus B = A \otimes I_b + I_a \otimes B$$

an affine function?

Also, if so, would the following function also be an affine transformation

$$f(A,B) := A \otimes M_1 + M_2 \otimes B$$

for any positive-definite matrices $$M_1$$ and $$M_2$$?

• What does it mean for a function of two arguments to be affine? Are you asking if the sum is affine with respect to each input? – Omnomnomnom Oct 8 '18 at 21:49
• I'm treating (A,B) as a vector, so asking if it is affine with respect to all the elements of both A and B. – Betterthan Kwora Oct 8 '18 at 23:13