# Matrices with complex entries

1. Oct 5, 2011

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1. The problem statement, all variables and given/known data
Let B be an m×n matrix with complex entries. Then by B* we denote the n×m matrix that is obtained by forming the transpose of B followed by taking the complex conjugate of each entry. For an n × n matrix A with complex entries, prove that if u*Au = 0 for all n × 1 column vectors u with complex entries, then A is a zero matrix.

3. The attempt at a solution
Let $$u = \left[ z_{1}, z_{2}, z_{3},..., z_{n} \right]^{T}$$

and $$u^{*} = \left[ \overline{z_{1}}, \overline{z_{2}}, \overline{z_{3}},...,\overline{z_{n}} \right]$$

Then
$$u^{*} A = \left[ {{\sum{\overline{z_{i}}a_{i1}}}},{{\sum{\overline{z_{i}}a_{i2}}}},...,{{\sum{\overline{z_{i}} a_{in}}}} \right]$$

And that's as far as I got, because I have no idea how to make $$a_{ij} = 0$$

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