(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This was a question in a recent exam and I would like to know if the answer I gave is correct since I not 100% sure...

If [itex]\mathbf{x}[/itex] and [itex]\mathbf{y}[/itex] are complex vectors in C^n (complex) and [itex]M[/itex] is a square (n x n)-matrix (also in C^n), prove that:

[tex]\mathbf{x} \cdot (M \mathbf{y}) = (M^* \mathbf{x}) \cdot \mathbf{y}[/tex]

(where M* denotes the complex conjugate + the transpose of M: [itex]M^* = \overline{M}^T[/itex]

(The dot denotes the standard complex dot-product)

3. The attempt at a solution

I did the following:

[tex]\mathbf{x} \cdot \mathbf{y} = \mathbf{x}^T \mathbf{\overline{y}}[/tex]

So

[tex]\mathbf{x} \cdot (M \mathbf{y}) = \mathbf{x}^T \overline{M \mathbf{y}} = \mathbf{x}^T \overline{M} \overline{\mathbf{y}}[/tex]

Now:

[tex]\mathbf{x}^T \overline{M} = (M^* \mathbf{x})^T[/tex] because [tex](M^* \mathbf{x})^T = \mathbf{x}^T (M^*)^T = \mathbf{x}^T \overline{M}[/tex]

So, now we have:

[tex]\mathbf{x} \cdot (M \mathbf{y}) = (M^* \mathbf{x})^T \overline{\mathbf{y}} = (M^* \mathbf{x}) \cdot \mathbf{y}[/tex]

Is this solution correct? Or did I make an error somewhere? (I'm not entirely sure of the very first statement for example...)

Thanks.

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# Proof x.(My) = (M*x).y

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