if $\mathrm{rank}(A)=m$ and $\mathrm{rank}(B)=n$, is that true that $\mathrm{rank}(A\otimes B)$ equal to $\mathrm{rank}(A)*\mathrm{rank}(B)$?

Question

Please help me... if I have $A,B$ are matrices and $\mathrm{rank}(A)=m$ and $\mathrm{rank}(B)=n$, is that true that $\mathrm{rank} (A\otimes B)$ equal to $\mathrm{rank}(A) * \mathrm{rank}(B)$? if false, what the true about $\mathrm{rank}(A\otimes B)$? thx a lot.

Answer 1

Yes it is true. Using the fact that $$ A\otimes B \cdot C\otimes D = AC \otimes BD $$ we can use row and column operations to put $A\otimes B$ into a convenient diagonal form.