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Chapter 3: Q20P (page 142)

Show that the determinant of a unitary matrix is a complex number with absolute value=1. Hint: See proof of equation (7.11).

Short Answer

Expert verified

The determinant of the unitary matrix is unity.

Step by step solution

01

Given information.

The matrix is unitary and the absolute value is 1.

02

Unitary matrix.

A unitary matrix is one whose conjugate transpose equals its inverse. Real orthogonal matrices have a complex analogue in the form of unitary matrices.

If Matrix A satisfies the condition AA†=A†A=I, then Matrix Ais a unitary matrix.

03

Show that the determinant of a unitary matrix is a complex number with absolute value of unity

Consider U is a unitary matrix.

U-1=UTUUT=IdetUUT=detIdet(U)detUT=detI(∵det(AB)=(detA)(detB))

Since,

detUT=detU-1detUT=det(U¯)detI=1

Then,(detU)(detU¯)=1

Now,

 zz¯=1      (∵det U=z)|z|2=1  |z|=1

Hence, from the above equation, it is clear that the determinant of the unitary matrix is unity.

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