Answer to Question #204180 in Linear Algebra for akur

Question #204180

2 Let A [ 1 𝑖 −𝑖 2 ] And let g be the form (on the space of 2x1 complex matrices) defined by g(X,Y) =Y*AX.Is g an inner product ? 


1
Expert's answer
2021-06-08T09:33:23-0400

Let X=(x1,x2),Y=(y1,y2)X=(x_1,x_2), Y=(y_1,y_2). Then g(X,Y)=YAX=x1yˉ1+2x2yˉ2+i(x2yˉ1x1yˉ2)g(X,Y)=Y^*AX=x_1\bar{y}_1+2x_2\bar{y}_2+i(x_2\bar{y}_1-x_1\bar{y}_2).

1) Check that g(X,Y)g(X,Y) is a sesquilinear form:

g(X+X,Y)=YA(X+X)=YAX+YAX=g(X,Y)+g(X,Y)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=YAX+YAX=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)=YA(cX)=cYAX=cg(X,Y)g(cX,Y)=Y^*A(cX)=cY^*AX=cg(X,Y),

g(X,cY)=(cY)AX=cˉYAX=cˉg(X,Y)g(X,cY)=(cY)^*AX=\bar{c}Y^*AX=\bar{c}g(X,Y).

2) Positivity: g(X,X)=x12+2x220g(X,X)=|x_1|^2+2|x_2|^2\geq 0 and g(X,X)>0g(X,X)>0 , if X(0,0)X\ne(0,0).

3) Symmetry: g(Y,X)=y1xˉ1+2y2xˉ2+i(y2xˉ1y1xˉ2)=g(X,Y)g(Y,X)=y_1\bar{x}_1+2y_2\bar{x}_2+i(y_2\bar{x}_1-y_1\bar{x}_2)=\overline{g(X,Y)}

Since all of the conditions to be a hermitean inner product are satisfied, we conclude that g(X,Y)g(X,Y) is a hermitean inner product on the space of 2x1 complex matrices.



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