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.
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1Are you asking about the Kronecker product of two matrices? – hardmath Jun 22 '15 at 04:09
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What was the point of introducing $m$ and $n$?? – Ben Grossmann Jun 22 '15 at 05:30
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This property can be deduced from the more general statement that the eigenvalues of the product are all values $\lambda \cdot \mu$ with $\lambda$ an eigenvalue of $A$ and $\mu$ an eigenvalue of $B$. – Ben Grossmann Jun 22 '15 at 05:33
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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.
TYS
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