By Alex Poznyak
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Extra info for Advanced Mathematical Tools for Control Engineers: Volume 1
I! (m − i)! 8. A matrix U ∈ Cn×n is unitary if and only if for any x, y ∈ Cn (U x, Uy) := (U x)∗ Uy = (x, y) Indeed, if U ∗ U = In×n then (U x, Uy) = (x, U ∗ Uy) = (x, y). Conversely, if (U x, Uy) = (x, y), then [U ∗ U − In×n ] x, y = 0 for any x, y ∈ Cn that proves the result. 9. If A and B are unitary, then AB is unitary too. Matrices and matrix operations 23 10. If A and B are normal and AB = BA (they commute), then AB is normal too. 11. If Ai are Hermitian (skew-Hermitian) and αi are any real numbers, then the matrix m αi Ai is Hermitian (skew-Hermitian) too.
If A¯ denotes the complex conjugate of A ∈ Cn×n , then det A¯ = det A Proof. Transforming det A¯ to the determinant of a triangular matrix triang A¯ and applying the rule ab = ab valid within the field C of complex values, we get n det A = det triang A = triang A ii i=1 n = (triang A)ii = det (triang A) = det A i=1 The result is proven. 6. 5. Let us consider the, so-called, n × n companion matrix ⎡ 0 1 ⎢ 0 0 ⎢ ⎢ · · Ca := ⎢ ⎢ · · ⎢ ⎣ 0 0 −a0 −a1 0 1 0 · · · 0 · 0 · · ⎤ · 0 · · ⎥ ⎥ · · ⎥ ⎥ · 0 ⎥ ⎥ 0 1 ⎦ · −an−1 associated with the vector a = (a0 , .
1. If A ∈ Rn×n , B ∈ Rm×m then 1. A ⊗ B = (A ⊗ In×n ) (Im×m ⊗ B) = (Im×m ⊗ B) (A ⊗ In×n ) (to prove this it is sufficient to take C = In×n and D = Im×m ). 2. (A1 ⊗ B1 ) (A2 ⊗ B2 ) · · · Ap ⊗ Bp = A1 A2 · · · Ap ⊗ B1 B2 · · · Bp for all matrices Ai ∈ Rn×n and Bi ∈ Rm×m (i = 1, . . , p). 11) Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 28 3. (A ⊗ B)−1 = A−1 ⊗ B −1 provided that both A−1 and B −1 exist. 2. If A ∈ Rn×n, B ∈ Rm×m then there exists a permutation P ∈ Rnm×nm such that P (A ⊗ B) P = B ⊗ A Proof.