Definite matrix: Difference between revisions

Revision as of 07:45, 2 June 2024

Revision as of 08:47, 2 June 2024

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then for any real vector <math>\mathbf{z}</math> with entries <math>a</math> and <math>b</math> we have <math>\mathbf{z}^\operatorname{T} 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</math> and <math>i</math>, one gets

<math display="block">\mathbf{z}^* M \mathbf{z} = \begin{bmatrix}1 & -i\end{bmatrix} M \begin{bmatrix}1 \\ i\end{bmatrix} = \begin{bmatrix}1 -+ i & -1 - i\end{bmatrix} \begin{bmatrix}1 \\ i\end{bmatrix} = 2-+2i</math>

which is not real. Therefore, <math>M</math> is not positive-definite.