# Matrix norm of Kronecker product

Is it true that $\| A \otimes B \| = \|A\|\|B\|$ for any matrix norm $\|\cdot \|$? If not, does this identity hold for matrix norms induced by $\ell_p$ vector norms?

• According to wikipedia, you can relate the eigenvalues of the Kronecker product to that of the operands. This should give you something for the spectral norm. Jun 30, 2017 at 17:40
• Yes, it's true for the spectral norm. That's the only case I know for certain.
– Rob
Jun 30, 2017 at 18:08
• Compute everything for two arbitrary $2\times 2$ matrices (i.e., get both sides as an expression of $a_{11}, a_{12},\dots ,b_{22}$). I'd expect counter examples to be easy to find once you have done that. Jun 30, 2017 at 18:12
• @rob in the answer below claims the statement is true, and hence there wont be counter examples Apr 2, 2022 at 13:47

Here is a proof for the lazy: Let $$A = \sum_i \sigma_i u_i v_i^T$$ and $$B = \sum_j \lambda_j x_j y_j^T$$ be the singular value decomposition (SVD) of the two matrices. Then, \begin{align} A \otimes B &= \sum_{i,j}\sigma_i \lambda_j (u_i v_i^T) \otimes (x_jy_j^T) \\ &= \sum_{i,j} \sigma_i \lambda_j (u_i \otimes x_j) (v_i^T \otimes y_j^T) \\ &= \sum_{i,j} \sigma_i \lambda_j (u_i \otimes x_j) (v_i \otimes y_j)^T \end{align} where the first equality is by the bilinearity of $$\otimes$$, the second by the "mixed product" property of Kronecker product and last one by $$(A\otimes B)^T = A^T \otimes B^T$$. It is not hard to see that $$\{u_i \otimes x_j, \forall i,j \}$$ is an orthonormal collection of vectors and similarly for $$\{v_i \otimes y_j, \forall i,j \}$$. It follows that the last line above is the SVD of $$A \otimes B$$, with singular values $$\{\sigma_i \lambda_j, \forall i,j\}$$. Hence, \begin{align} \| A \otimes B\| = \max_{i,j} \sigma_i \lambda_j = (\max_i \sigma_i)(\max_j \lambda_j) = \|A\| \|B\|. \end{align} for the $$\ell_2$$ operator norm.
Theorem 8 here provides the answer: http://www.ams.org/journals/mcom/1972-26-118/S0025-5718-1972-0305099-X/S0025-5718-1972-0305099-X.pdf. As discussed on page 413, the identity holds for all matrix norms induced by $\ell_p$ vector norms. In fact, it seems to hold for any induced vector norm, or any submultiplicative norm.