**12.** Let *A* ϵ *M*_{m,n} and *B* ϵ *M*_{p,q} be given. Use Corollary (4.3.10) to show that *A ⊗ B* and *B ⊗ A* have the same singular values.

**13.** Let *A* ϵ *M*_{n} and *B* ϵ *M*_{m} be given square matrices. Use Corollary (4.3.10) to show that [counting multiplicities in (e) and (f)]

(a) det(*A ⊗ B*) = det(*B ⊗ A*).

(b) tr(*A ⊗ B*) = tr(*B ⊗ A*).

(c) If *A ⊗ B* is normal, then so is *B ⊗ A*.

(d) If *A ⊗ B* is unitary, then so is *B ⊗ A*.

(e) The eigenvalues of *A ⊗ B* are the same as those of *B ⊗ A*.

(f) The singular values of *A ⊗ B* are the same as those of *B ⊗ A*.

**14.** Use the mixed-product property (4.2.10) to prove the assertions in (b), (c), and (d) of Theorem (4.3.17). Unfortunately, it does not seem to be possible to prove (a) so easily.

**15.** Let

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