Question

prove for any matrix A , A+(A)^theta is hermitian matrix ​

Answer 1

Answer:

By definition, A∗A=AT¯¯¯¯¯¯¯A

Then (A∗A)∗=(AT¯¯¯¯¯¯¯A)T¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯=ATA¯¯¯¯¯¯¯¯¯¯¯¯¯¯=AT¯¯¯¯¯¯¯A=A∗A. So A∗A is Hermitian.

Let x be any vector, then

xT¯AT¯¯¯¯¯¯¯Ax=(Ax)T¯¯¯¯¯¯¯¯¯¯¯¯¯Ax=∥Ax∥2⩾0

So AT¯¯¯¯¯¯¯A or A∗A is semi-positive definite matrix and thus has non-negative eigenvalues.

Same reasoning applies for AA∗