Definite matrix: Difference between revisions

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The definition of positive definite can be generalized by designating any complex matrix <math>M</math> (e.g. real non-symmetric) as positive definite if <math>\mathcal{R_e} \left\{\mathbf{z}^* M \mathbf{z}\right\} > 0</math> for all non-zero complex vectors <math>\mathbf{z},</math> where <math>\mathcal{R_e}\{c\}</math> denotes the real part of a [[complex number]] <math>c ~.</math><ref name="mathw">{{cite web |last = Weisstein |first = Eric W. |url = http://mathworld.wolfram.com/PositiveDefiniteMatrix.html |title = Positive definite matrix |website = MathWorld |publisher = Wolfram Research |access-date= 2012-07-26 }}</ref> Only the Hermitian part <math display="inline">\frac{1}{2}\left(M + M^*\right)</math> determines whether the matrix is positive definite, and is assessed in the narrower sense above. Similarly, if <math>\mathbf{x}</math> and <math>M</math> are real, we have <math>\mathbf{x}^\top M \mathbf{x} > 0</math> for all real nonzero vectors <math>\mathbf{x}</math> if and only if the symmetric part <math display="inline">\frac{1}{2}\left(M + M^\top \right)</math> is positive definite in the narrower sense. It is immediately clear that <math display="inline">\mathbf{x}^\top M \mathbf{x} = \sum_{ij} x_i M_{ij} x_j</math>is insensitive to transposition of <math>M ~.</math>

Consequently, aA non-symmetric real matrix with only positive eigenvalues doesmay nothave needa tosymmetric part with negative eigenvalues, in which case it will not be positive (semi)definite. For example, the matrix <math display=inline>M = \left[\begin{matrix} 4 & 9 \\ 1 & 4 \end{matrix}\right]</math> has positive eigenvalues yet1 isand not7, positive definite; in particular a negative value ofyet <math>\mathbf{x}^\top M \mathbf{x} = -2 </math> is obtained with the choice <math>\mathbf{x} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right] </math> (which is the eigenvector associated with the negative eigenvalue of the symmetric part of {{nowrap|<math>M</math>).}}

In summary, the distinguishing feature between the real and complex case is that, a [[Bounded operator|bounded]] positive operator on a complex Hilbert space is necessarily Hermitian, or self adjoint. The general claim can be argued using the [[polarization identity]]. That is no longer true in the real case.