Question

Show the product of two unitary matrices is unitary

Answer 1

unitary always matrices are square right?

Answer 2

\[\text{Unitary:}\textbf U ^{\dagger}= \textbf{U}^{-1}\]

Answer 3

\[\textbf {U W}=\]

Answer 4

you can only take the product if they are the same size

Answer 5

Assume that A and B are both unitary matrices. Then, AB(AB)* = AB(B*A*) = A(BB*)A* = AIA* = AA* = I. Thus AB(AB)* = I, which is the definition of a unitary matrix, which implies that AB is unitary. * is the conjugate transpose of a matrix.

Answer 6

Because if the product is unitary, then it must satisfy AB(AB)* = I

Answer 7

simpler than i was imagining, thanks again bmp

Answer 8

No problem :-)

Answer 9

\[ \textbf {W}=\textbf {UV} \]\[\textbf {W} ^\dagger=(\textbf{UV})^\dagger \]\[=\textbf{V}^\dagger \textbf{U}^\dagger\]\[=\textbf V ^{-1} \textbf U^{-1}\]\[=(\textbf{UV})^{-1} \]\[\textbf{W}^\dagger=\textbf {W}^{-1}\]