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Rodrigo de Azevedo
Rodrigo de Azevedo
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Conditons Conditions on symmetric 3X3$3 \times 3$ matrices to satisfy the convex equality for cofactor and determinant

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Daniele Tampieri
Daniele Tampieri
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Given any 3X3$3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such that:

$\cof(\sum_{i=1}^na_i(A_i+X_i))=\sum_{i=1}^na_i\cof (A_i +X_i)$$\DeclareMathOperator{\cof}{cof}$ $$ \cof\left(\sum_{i=1}^na_i(A_i+X_i)\right)=\sum_{i=1}^na_i\cof (A_i +X_i) $$ and $\det(\sum_{i=1}^na_i(A_i+X_i))=\sum_{i=1}^na_i\det (A_i+X_i).$

I$$ \det\left(\sum_{i=1}^na_i(A_i+X_i)\right )=\sum_{i=1}^na_i\det (A_i+X_i). $$ I know that having $\cof(\sum_{i=1}^na_iA_i)-\sum_{i=1}^na_i\cofA_i $$\cof\left(\sum_{i=1}^na_iA_i\right)-\sum_{i=1}^na_i\cof A_i $ positive semidefinite is a necessary condition. Is it sufficient?? Or which condition we should add!?

Given any 3X3 finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such that:

$\cof(\sum_{i=1}^na_i(A_i+X_i))=\sum_{i=1}^na_i\cof (A_i +X_i)$ and $\det(\sum_{i=1}^na_i(A_i+X_i))=\sum_{i=1}^na_i\det (A_i+X_i).$

I know that having $\cof(\sum_{i=1}^na_iA_i)-\sum_{i=1}^na_i\cofA_i $ positive semidefinite is a necessary condition. Is it sufficient?? Or which condition we should add!

Given any $3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such that:$\DeclareMathOperator{\cof}{cof}$ $$ \cof\left(\sum_{i=1}^na_i(A_i+X_i)\right)=\sum_{i=1}^na_i\cof (A_i +X_i) $$ and $$ \det\left(\sum_{i=1}^na_i(A_i+X_i)\right )=\sum_{i=1}^na_i\det (A_i+X_i). $$ I know that having $\cof\left(\sum_{i=1}^na_iA_i\right)-\sum_{i=1}^na_i\cof A_i $ positive semidefinite is a necessary condition. Is it sufficient?? Or which condition we should add?

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user519646
user519646
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Conditons on symmetric 3X3 matrices to satisfy the convex equality for cofactor and determinant

Given any 3X3 finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such that:

$\cof(\sum_{i=1}^na_i(A_i+X_i))=\sum_{i=1}^na_i\cof (A_i +X_i)$ and $\det(\sum_{i=1}^na_i(A_i+X_i))=\sum_{i=1}^na_i\det (A_i+X_i).$

I know that having $\cof(\sum_{i=1}^na_iA_i)-\sum_{i=1}^na_i\cofA_i $ positive semidefinite is a necessary condition. Is it sufficient?? Or which condition we should add!