Page 90 - Textos de Matemática Vol. 36
P. 90
80 Appendix A. “stacking” the columns of X into one long vector of dimension mn is called the
vec-function of X and it is written X⃗ or vec X. m×nmn1n ⃗x2
x1 vec:C →C , X=(x,...,x)→vec(X)=X:= . .
. xn
Remark A.1.2. Let A, A1, A2 ∈ Cn×n, B, B1, B2 ∈ Cm×m. Then the following statements all hold:
(i) (A1A2) ⊗ (B1B2) = (A1 ⊗ B1)(A2 ⊗ B2).
(ii) (A ⊗ B)−1 = A−1 ⊗ B−1 if A and B are invertible.
(iii) If A has the eigenvalues λ1,...,λn and B has the eigenvalues μ1,...,μm, then A ⊗ B has the nm eigenvalues
λjμk, 1≤j≤n, 1≤k≤m, and (A ⊗ Im) + (In ⊗ B) has the nm eigenvalues
λj +μk, 1≤j≤n, 1≤k≤m.
(iv) For S ∈ Cn×m,
vec (ASB) = (BT ⊗ A)S⃗; consequently the linear matrix equation
A1XB1 + A2XB2 = C has the equivalent representation
[(B1T ⊗A1)+(B2T ⊗A2)]X⃗ =C⃗.
For more details, see [LaTi85].
Using the results stated in Remark A.1.2 we can rewrite algebraic and differential Sylvester and Lyapunov equations as standard linear equations and apply the classical theory for linear algebraic and differential equations in order to prove the following results.
