Question

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 ? 

Answer 1

Let $X=(x_1,x_2), Y=(y_1,y_2)$. Then $g(X,Y)=Y^*AX=x_1\bar{y}_1+2x_2\bar{y}_2+i(x_2\bar{y}_1-x_1\bar{y}_2)$.

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

$g(X+X',Y)=Y^*A(X+X')=Y^*AX+Y^*AX'=g(X,Y)+g(X',Y)$

$g(X,Y+Y')=(Y^*+Y'^*)AX=Y^*AX+Y'^*AX=g(X,Y)+g(X,Y')$

$g(cX,Y)=Y^*A(cX)=cY^*AX=cg(X,Y)$,

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

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

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

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