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

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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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