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I've been trying to rewrite the matrix $\left( \begin{array}{cccc} x_{n} & x_{1} & x_{2} & \cdots & x_{n-1} \\ x_{1} & x_{n} & 0 &\cdots & 0 \\ x_{2} & 0& \ddots & 0 & \vdots\\ \vdots & \vdots& & \ddots \\ x_{n-1} & 0 & \cdots & & x_{n} \end{array} \right) $

into the form $-A_{0}+\sum_{k=1}^{n}A_{k}x_{k}$ for matrices $A_{k}$ and scalars $x_{k}$.

Could anyone help me with this?

The main problem I have is that $\sum_{k=1}^{n}A_{k}x_{k}$ gives a vector and im suppose to add this to the matrix $-A_{0}$? How does one add a vector to a matrix in this context?

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    My linear algebra is not excellent but I do understand that which is the main reason why I'm so confused. I haven't thought of it myself btw. $-A_{0}+\sum A_{k}x_{k}$ is the constraint for the primal semidefinite program in convex optimization. I am simply trying to find someone who can help me interprete this..2012-11-11

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Call $A$ your matrix, and, for every $1\leqslant i\leqslant n$, $A_i$ the matrix $A$ when every $x_k$ is zero except $x_i=1$ (for example, $A_n=I$). Then $A=\sum\limits_{i=1}^nx_iA_i$ (hence there is no $A_0$).

Note that $A$ and every $A_i$ are $n\times n$ matrices while every $x_i$ is a scalar. Recall that if $y$ is a scalar and $B=(B_{kj})_{kj}$ a matrix, then $yB$ is the matrix of the same size as $B$ with entries $(yB_{kj})_{kj}$.

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    Ah apologies $x$ is the vector with the entries $x_{k}$. It all makes a lot more sense now..2012-11-11