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Matrices with complex entries

  1. Oct 5, 2011 #1
    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 [tex]u = \left[ z_{1}, z_{2}, z_{3},..., z_{n} \right]^{T}[/tex]

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

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

    And that's as far as I got, because I have no idea how to make [tex]a_{ij} = 0[/tex]
     
  2. jcsd
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