- #1

- 102

- 0

Definition: Let A[itex]\in[/itex]M

_{n}(ℂ) the complex vector space

C(A)={X[itex]\in[/itex]M

_{n}(ℂ) : XA=AX}

For A[itex]\in[/itex]M

_{n}(ℂ) which is similar to A* we define the complex vector spaces:

C(A,A*)={S[itex]\in[/itex]M

_{n}(ℂ) : SA=A*S}

H(A,A*)={H[itex]\in[/itex]M

_{n}(ℂ): H is Hermitian and HA=A*H} [itex]\subset[/itex] C(A,A*)

Define a map T:C(A,A*)→H(A,A*) by T(S)=[itex]\frac{1}{2}[/itex]S + [itex]\frac{1}{2}[/itex]S*

As a map between real vector spaces, T is linear and Kern(T)={X[itex]\in[/itex]M

_{n}: X is skew Hermitian}=iH(A,A*)

I just want to make sure that my understanding is correct and what is "Kern" short for

To say that P[itex]\in[/itex]Kern(T) means that P is an element of C(A,A*) which means that PA=A*P such that P is skew Hermitian

the defintion is from the paper I am reading it is by J. Vermeer on page 263

http://www.math.technion.ac.il/iic/ela//ela-articles/articles/vol17_pp258-283.pdf

Thank you for any further comments