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12.    Let A ϵ Mm,n and B ϵ Mp,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 Mn and B ϵ Mm 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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