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Dv's question at Yahoo! Answers (Hermitian matrix)

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Here is the question:

A and B are 2 matrices. Normally I use a dagger symbol for the Hermitian conjugate; here I'll use a £ sign as I have no dagger symbol.

If I have a matrix (AB+BA), then take the hermitian conjugte (AB + BA)£, does that mean:

(AB+BA)£=AB£+BA£

So, if A and B are hermitian, then (AB-BA) and (AB+BA) are also hermitian, right?
Here is a link to the question:

Hermitian matrix question? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Hello Dv,

For all $A,B\in \mathbb{C^{n\times n}}$ hermitian matrices, and using well known properties of the ${}^\dagger$ operator: $$\begin{aligned}(AB+BA)^\dagger&=(AB)^\dagger+(BA)^\dagger\\&=B^\dagger A^\dagger +A^\dagger B^\dagger\\&=BA+AB\\&=AB+BA\\&\Rightarrow AB+BA\mbox{ is hermitian}\end{aligned}$$ $$\begin{aligned}(AB-BA)^\dagger&=(AB)^\dagger-(BA)^\dagger\\&=B^\dagger A^\dagger -A^\dagger B^\dagger\\&=BA-AB\\&=-(AB-BA)\\&\Rightarrow AB-BA\mbox{ is skew-hermitian}\end{aligned}$$ If you have further questions, you can post them in the Linear and Abstract Algebra section.