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By this definition, a positive-definite ''real'' matrix <math>M</math> is Hermitian, hence symmetric; and <math>\mathbf{z}^\top M\mathbf{z}</math> is positive for all non-zero ''real'' column vectors <math>\mathbf{z} .</math> However the last condition alone is not sufficient for <math>M</math> to be positive-definite. For example, if
<math display="block">M = \begin{bmatrix} ~1~ & ~1~ \\-1~ & ~1~ \end{bmatrix},</math>
then for any real vector <math>\mathbf{z}</math> with entries <math>a</math> and <math>b</math> we have <math>\mathbf{z}^\top M\mathbf{z} = \left(a + b\right)a + \left(-a + b\right) b = a^2 + b^2,</math> which is always positive if <math>\mathbf{z}</math> is not zero. However, if <math>\mathbf{z}</math> is the complex vector with entries {{math|1}} and <math>i,</math> one gets
<math display="block">\mathbf{z}^* M\mathbf{z} = \begin{bmatrix}
which is not real. Therefore, <math>M</math> is not positive-definite.
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