Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let a, b, c, d ∈ C and consider the vector space $C^2$

Suppose inner product is defined as:

$⟨x, y⟩ = ax_1\bar y_1 + bx_2\bar y_1 + cx_1\bar y_2 + dx_2\bar y_2$

I am trying to find all a, b, c, d such that the definition above is false.

I believe I checked all the points in definition of inner product over complex field and found only

a = b = c = d = 0

as it contradicts $⟨x, x⟩ = 0 \implies x=0 $

Can you find anything else?

share|improve this question
    
What happens when $b=c=d=0$ and $a\neq 0$? –  Brad Apr 7 at 17:17
    
in this case ⟨x,x⟩ = 0 iff x = 0 –  tmac_balla Apr 7 at 18:08
    
Can you prove that? –  Brad Apr 7 at 21:13

1 Answer 1

Let $$M=\begin{pmatrix}a&b\\c&d\end{pmatrix}$$

You have$$\langle x,y\rangle=xMy^*$$

You want $\langle x,x\rangle=0\implies x =0$. Suppose that $M$ isn't invertible. Take any $v\in\operatorname{Ker}(M)\setminus\{0\}$. $v\not=0$ and $v^*Mv=0$ so $M$ does not represent a scalar product. You therefore need $M$ to be invertible for it to represent a scalar product.

So taking any matrix $M$ with $\det M = 0$ would give you a counterexample.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.