- #1
Nick89
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Homework Statement
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)
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.