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Let $x_1,\dots, x_n$ be a set of $d-$dimensional real vectors. Consider the sum of rank-one matrices given as \begin{align}K = \sum_{i,j}(x_i^Tx_j)*(x_ix_j^T)\end{align} Here $x_i^Tx_j$ is the dot-product of vectors $x_i$ and $x_j$. And $x_ix_j^T$ denotes the outer product of the same vectors. Is there a nice way of writing this in terms of the matrix $X=[x_1,\dots,x_n]$ (i.e. columns are $x_i$). Are there any interesting properties for $K$?. Can $K$ be positive definite?

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We can write your matrix $K$ as $$ K = \sum_{i,j = 1} x_ix_i^Tx_jx_j^T = \left(\sum_i x_ix_i^T\right) \left(\sum_j x_jx_j^T\right) = \left(\sum_i x_ix_i^T\right)^2 = (XX^T)^2 $$ Your matrix $K$ will always be positive semidefinite. It will be positive definite if and only if the columns of $X$ span all of $\Bbb R^d$.

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