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Added the hypothesis of semidefiniteness.
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Vamsi
Vamsi
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Let $f : H_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian, positive semidefinite matrices $f(A) = \frac{M_i (A)}{\det(A)}$ where $M_i(A)$ is the determinant of the the submatrix formed by the $A_{kl}$ where $1\leq k, l \leq i$, i.e., it is the $i\times i$ principal minor of $A$. Is the function $f$ convex ?

Let $f : H_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian matrices $f(A) = \frac{M_i (A)}{\det(A)}$ where $M_i(A)$ is the determinant of the the submatrix formed by the $A_{kl}$ where $1\leq k, l \leq i$, i.e., it is the $i\times i$ principal minor of $A$. Is the function $f$ convex ?

Let $f : H_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian, positive semidefinite matrices $f(A) = \frac{M_i (A)}{\det(A)}$ where $M_i(A)$ is the determinant of the the submatrix formed by the $A_{kl}$ where $1\leq k, l \leq i$, i.e., it is the $i\times i$ principal minor of $A$. Is the function $f$ convex ?

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Vamsi
Vamsi
  • 3.4k
  • 25
  • 38

Convexity of a (non-symmetric) function of matrices

Let $f : H_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian matrices $f(A) = \frac{M_i (A)}{\det(A)}$ where $M_i(A)$ is the determinant of the the submatrix formed by the $A_{kl}$ where $1\leq k, l \leq i$, i.e., it is the $i\times i$ principal minor of $A$. Is the function $f$ convex ?