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oc.optimization and management – Positivity of quadratic figure minus linear figure on the simplex

Let us $a_{ij}$ breathe the weather of a n dimensional covariance matrix. Can we show that:

$ 1-sum_{ok=1}^n a_{ik} lambda_k + sum_{j=1}^n sum_{ok=1}^n lambda_j a_{jk} lambda_k >0$

for $i=1 ldots n$

the place the lambdas are constrained by: $ sum_{ok=1}^n lambda_k = 1$ and $lambda_i > 0$ for $i=1ldots n$, Or in a extra common route, what are the circumstances that the weather of the covariance matrix ought to answer in order that the above clique of inequalities maintain?

NOTE: in matrix figure, if $A$ is a covariance matrix and $a_i$ is a row vector having the i-th row of $A$, the query is:

is $1-a_i lambda + lambda^T A lambda >0 $ the place $lambda=[lambda_1 ldots lambda_n]^T$ and $lambda^T e_n $ =1 with $lambda_k >0$ and $e_n$ a column vector of ones.

The query ought to breathe formulated as: discover the circumstances for the weather of matrix $A$ in order that the inequality holds for $i=1,ldots n$.

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