Let X=(x1,x2),Y=(y1,y2). Then g(X,Y)=Y∗AX=x1yˉ1+2x2yˉ2+i(x2yˉ1−x1yˉ2).
1) Check that g(X,Y) is a sesquilinear form:
g(X+X′,Y)=Y∗A(X+X′)=Y∗AX+Y∗AX′=g(X,Y)+g(X′,Y)
g(X,Y+Y′)=(Y∗+Y′∗)AX=Y∗AX+Y′∗AX=g(X,Y)+g(X,Y′)
g(cX,Y)=Y∗A(cX)=cY∗AX=cg(X,Y),
g(X,cY)=(cY)∗AX=cˉY∗AX=cˉg(X,Y).
2) Positivity: g(X,X)=∣x1∣2+2∣x2∣2≥0 and g(X,X)>0 , if X=(0,0).
3) Symmetry: g(Y,X)=y1xˉ1+2y2xˉ2+i(y2xˉ1−y1xˉ2)=g(X,Y)
Since all of the conditions to be a hermitean inner product are satisfied, we conclude that g(X,Y) is a hermitean inner product on the space of 2x1 complex matrices.
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