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I have symmetric matrix which is formed by complex integer vectors as follows \begin{bmatrix} \|f_1\|^2 & af_2^Hf_1 & bf_3^Hf_1 & \dots \\ a^*f_1^Hf_2 & \|f_2\|^2 & cf_3^Hf_2 & \dots \\ b^*f_1^Hf_3 & c^*f_2^Hf_3 & \|f_3\|^2 & \dots \\ \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix} where $f_i$s are complex integer vectors and $a, b ,c $ are complex numbers between 0 to 1. I want to show that if I replace the off-diagonal elements by the norm of the vectors the determinant of the resulted matrix becomes smaller compared to the original matrix. Please note that the off-diagonal elements are increased. \begin{bmatrix} \|f_1\|^2 & a\|f_2\|\|f_1\| & b\|f_3\|\|f_1\| & \dots \\ a^*\|f_1\|\|f_2\| & \|f_2\|^2 & c\|f_3\|\|f_2\| & \dots \\ b^*\|f_1\|\|f_3\| & c^*\|f_2\|\|f_3\| & \|f_3\|^2 & \dots \\ \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix}

mehrdad
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  • What is $f^H g$? Is it the Hermitian product of $f$ and $g$? And what do you mean by "complex integer vector"? – Alex M. May 07 '16 at 18:47
  • yes,$ f^Hg$ is the Hermitian product of $f$ and $g$ and by complex integer vector I mean a vector with entries $m+j*n$, where $m$ and $n$ are integer numbers. – mehrdad May 08 '16 at 01:45

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