Demonstrate the following facts about the Kronecker product of two matrices:

(a) c(A ⊗ B)=(cA) ⊗ B = A ⊗ (cB) for any scalar c.

(b) (A ⊗ B)t = At ⊗ Bt.

(c) (A + B) ⊗ C = A ⊗ C + B ⊗ C.

(d) A ⊗ (B + C) = A ⊗ B + A ⊗ C.

(e) (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C).

(f) (A ⊗ B)(C ⊗ D)=(AC) ⊗ (BD).

(g) If A and B are invertible square matrices, then

(A ⊗ B)

−1 = A−1 ⊗ B−1.

(h) If λ is an eigenvalue of the square matrix A with algebraic multiplicity r and µ is an eigenvalue of the square matrix B with algebraic multiplicity s, then λµ is an eigenvalue of A ⊗ B with algebraic multiplicity rs.

(i) If A and B are square matrices, tr(A ⊗ B) = tr(A) tr(B).

(j) If A is an m × m matrix, and B is an n × n matrix, then det(A ⊗ B) = det(A)

n det(B)

m.

All asserted operations involve matrices of compatible dimensions.

(Hint: For part (h), let A = USU −1 and B = VTV −1 be the Jordan canonical forms of A and B. Check that S ⊗ T is upper triangular.)