Does \(((x_1, x_2) \mid (y_1, y_2)) = x_1\bar{y}_1 + 2x_2\bar{y}_1 + 2x_1\bar{y}_2 + x_2\bar{y}_2\) define an inner product on \(C^2\)?
Solution
No. The form can be written as \[ (X\mid Y)=Y^*AX, \qquad A= \begin{pmatrix} 1&2\\ 2&1 \end{pmatrix}. \] The matrix \(A\) is Hermitian, so the form has the required conjugate symmetry. However it is not positive definite.
Take \[ X=\binom1{-1}. \] Then \[ AX=\binom{-1}{1}, \] and therefore \[ (X\mid X) =X^*AX =(1,-1)\binom{-1}{1} =-2<0. \] An inner product must satisfy \((X\mid X)>0\) for every nonzero \(X\). Hence the displayed form does not define an inner product on \(\mathbb C^2\).