= Positive diagonal symmetrization of a matrix
If a real <matrix> $B$ has a positive <diagonal matrix> $W$ such that $WB$ is a <symmetric matrix>, then $H=W^{1/2}BW^{-1/2}$ is a <symmetric matrix> related to $B$ by a <similarity transformation>: $B^TW=WB$ gives $H^T=H$. Thus $B$ is a <diagonalizable matrix> with real <eigenvalues>. If $\mathbf z^TWB\mathbf z\le0$ for all real <vectors> $\mathbf z$, its <eigenvalues> are nonpositive. This converts symmetry in a weighted <inner product> into Euclidean symmetry.
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