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Euge
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Let ##M## be a real ##n \times n## matrix. If ##M + M^T## is positive definite, show that $$\det\left(\frac{M + M^T}{2}\right) \le \det M$$
The Cholesky decomposition was my idea, too, but I do not see the induction part. "All these are non-negative" appears to be wrong to me, at least I am not convinced. Applied to the two by two case ##\begin{pmatrix}a&b\\c&d\end{pmatrix}##, it comes down to ##(c-b)^2>0## which does not say anything about ##b## and ##c##, and higher dimensions get even messier. The induction has to be executed in detail, and not by "All these are non-negative".Hill said:A positive definite matrix can be diagonalized. This transformation, when applied to ##M## and ##M^T##, keeps them transpose of each other. Since now their sum is a diagonal matrix, they are antisymmetric with diagonal. So, we need to show that $$\det D_n \le \det (A_n+D_n)$$
This can be shown by induction. ##\det D_n## is product of its diagonal elements. From the Leibniz formula, we see that the terms of ##\det (A_n+D_n)## contain the terms of ##\det A_n## plus various products of diagonal elements and ##\det A_m## for m<n. All these are non-negative, and one of the latter is the product of all the diagonal elements. Thus, their sum is greater or equal to the product of all the diagonal elements. ##\blacksquare##
Sorry, once on the wrong track (Cholesky) required a wagon-by-wagon help to get back on the right one.Hill said:I do not use Cholesky decomposition, but eigenvalue decomposition of the matrix ##S=\frac {M+M^T} 2##. If I am not mistaken, the real symmetric positive definite matrix can be diagonalized. Let ##S=PDP^{-1}##. Then, $$D=P^{-1}SP=\frac {P^{-1}MP+P^{-1}M^TP} 2=\frac {B+B^T} 2$$
This implies that ##B## has the same diagonal elements as ##D##, and all other elements in ##B## are antisymmetric.
Would you please refer me to where I can read about this kind of decomposition? Thanks.julian said:Since MS is positive definite there exists MS12 such that MS=MS12MS12.
After diangonalization of ##M_s## which has all positive eigenvalues ##\{\lambda_i\}##, ##M^{1/2}_s## is also a diangonal matrix whose eigenvalues are ##\{\sqrt{\lambda}_i\}##.Hill said:Would you please refer me to where I can read about this kind of decomposition? Thanks.
P.S. I can prove this specific one to myself, but maybe there is more to it.