Hello, all! Could somebody draw a proof-sketch of next expression from tensor algebra on matrices over finite fields: determinant of tensor product $A~ \times ~B$ of $n \times n$-matrix $A$ over finite field $GF(q)$ on $m \times m$-matrix $B$ over finite field $GF(q)$ is $\det(A)^m \cdot \det(B)^n$.

Please, give me a link or reference if it is online or in some book. Thank you.

  • $\begingroup$ Would that there was a wikipedia equivalent to lmgtfy.com... en.wikipedia.org/wiki/Kronecker_product $\endgroup$ – Ketil Tveiten Mar 1 '11 at 12:56
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    $\begingroup$ Hint: $A\otimes B = \left(A\otimes I_m\right)\circ \left(I_n\otimes B\right)$. $\endgroup$ – darij grinberg Mar 1 '11 at 13:01
  • $\begingroup$ Excuse me, please. I am sorry, but I forgot to write that I need this fact satisfaction in case of finite field $\endgroup$ – spk Mar 1 '11 at 13:02
  • $\begingroup$ The argument given on the Wikipedia is the standard ugly one. $\endgroup$ – darij grinberg Mar 1 '11 at 13:02
  • $\begingroup$ spk: It doesn't matter which field or ring you are in (as long as it is commutative). $\endgroup$ – darij grinberg Mar 1 '11 at 13:03

Darij's first comment could be made into an answer as follows.

Darij advised to write

$$A \otimes B = (A \circ I_n) \otimes (I_m \circ B) = (A \otimes I_m) \circ (I_n \otimes B)$$

where the second equation follows from functoriality of the tensor product. Here both $A \otimes I_m$ and $I_n \otimes B$ are square matrices of size $m n \times m n$. Since the determinant from such matrices to the scalar field is a monoid homomorphism, the determinant of the last expression is

$$\det(A \otimes I_m) \det(I_n \otimes B)$$

so we are left to determine the two determinants above. Since these are similar, we do the first. We may express an $m$-dimensional vector space $k^m$ as a direct sum of 1-dimensional vector spaces, so

$$A \otimes I_{k^m} = A \otimes (I_k \oplus \ldots \oplus I_k) = (A \otimes I_k) \oplus \ldots \oplus (A \otimes I_k)$$

because tensor products preserve direct sums. This is just $A \oplus \ldots \oplus A$. This matrix consists of $m$ blocks of $A$, so its determinant is $\det(A)^m$, and we are done.

  • $\begingroup$ Thank you for your assistance. It is what I really need with my question $\endgroup$ – spk Mar 1 '11 at 14:34

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