Thread: complex conjugates and unitary

1. complex conjugates and unitary

For M$\displaystyle \in$M_nxn(C), let the $\displaystyle \overline{M}$ be the matrix such that $\displaystyle \overline{(M)}$_ij=$\displaystyle \overline{M}$ _ij for all i,j, where $\displaystyle \overline{M}$_ij is the complex conjugate of M_ij.

(1) Prove that det$\displaystyle \overline{M}$ =$\displaystyle \overline{det(M)}$

(2) A matrix Q in M_nxn(C)is called unitary if QQ*=I, where Q*=$\displaystyle \overline{Q^t}$. Prove that if Q is a unitary matrix, then |det(Q)|=1

2. Originally Posted by tn11631
For M$\displaystyle \in$M_nxn(C), let the $\displaystyle \overline{M}$ be the matrix such that $\displaystyle \overline{(M)}$_ij=$\displaystyle \overline{M}$ _ij for all i,j, where $\displaystyle \overline{M}$_ij is the complex conjugate of M_ij.

(1) Prove that det$\displaystyle \overline{M}$ =$\displaystyle \overline{det(M)}$

This follows at once from the definition of determinant and because complex conjugation is and additive and multiplicative function

(2) A matrix Q in M_nxn(C)is called unitary if QQ*=I, where Q*=$\displaystyle \overline{Q^t}$. Prove that if Q is a unitary matrix, then |det(Q)|=1
As $\displaystyle \det Q^*=\overline{\det Q}$ ,as was proved in (1), you get $\displaystyle \det I=\det (QQ^*)=\det Q\cdot\det Q^*$ ...etc.

Tonio