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BrainHurts
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I need some help understanding the following definition:
Definition: Let A[itex]\in[/itex]Mn(ℂ) the complex vector space
C(A)={X[itex]\in[/itex]Mn(ℂ) : XA=AX}
For A[itex]\in[/itex]Mn(ℂ) which is similar to A* we define the complex vector spaces:
C(A,A*)={S[itex]\in[/itex]Mn(ℂ) : SA=A*S}
H(A,A*)={H[itex]\in[/itex]Mn(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]Mn: 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
Definition: Let A[itex]\in[/itex]Mn(ℂ) the complex vector space
C(A)={X[itex]\in[/itex]Mn(ℂ) : XA=AX}
For A[itex]\in[/itex]Mn(ℂ) which is similar to A* we define the complex vector spaces:
C(A,A*)={S[itex]\in[/itex]Mn(ℂ) : SA=A*S}
H(A,A*)={H[itex]\in[/itex]Mn(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]Mn: 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