Editing Definite matrix (section)

=== Consistency between real and complex definitions ===
Since every real matrix is also a complex matrix, the definitions of "definiteness" for the two classes must agree.

For complex matrices, the most common definition says that <math>M</math> is positive-definite if and only if <math>\mathbf{z}^* M\mathbf{z}</math> is real and positive for every non-zero complex column vectors <math>\mathbf{z} .</math>  This condition implies that <math>M</math> is Hermitian (i.e. its transpose is equal to its conjugate), since <math>\mathbf{z}^* M\mathbf{z}</math> being real, it equals its conjugate transpose <math>\mathbf{z}^*M^*\mathbf{z}</math> for every <math>\mathbf{z},</math> which implies <math>M = M^* .</math> 

By this definition, a positive-definite ''real'' matrix <math>M</math> is Hermitian, hence symmetric; and <math>\mathbf{z}^\mathsf{T} 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}^\mathsf{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}} and {{tmath| i }}, 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.

On the other hand, for a ''symmetric'' real matrix <math>M,</math> the condition "<math>\mathbf{z}^\mathsf{T} M\mathbf{z} > 0</math> for all nonzero real vectors <math>\mathbf{z}</math>" ''does'' imply that <math>M</math> is positive-definite in the complex sense.