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\end{matrix}\right)$end{smallmatrix}\right)$?

Edit Thank you Greg for your answer. In our comments, we observe that every psd matrix of the form $\left(\begin{smallmatrix}a & b & c & d\\b & c & d & e\\c & d & e & f\\d & e & f & g\end{smallmatrix}\right)$ is a finite sum of rank 1 and rank 2 matrices. Can one prove the following in a convex analytic manner?

A psd matrix of the form $\left(\begin{smallmatrix} a & b & c & d\\ b & c & d & e\\ c & d & e & f\\ d & e & f & g \end{smallmatrix}\right)$ can be written as the sum of finitely many rank 1 matrices of the form $\left(\begin{smallmatrix} x^6 & x^5 y & x^4 y^2 & x^3 y^3\\ x^5 y & x^4 y^2 & x^3 y^3 & x^2 y^4\\ x^4 y^2 & x^3 y^3 & x^2y^4 & x y^5 \\ x^3y^3 & x^2y^4 & x y^5 & y^6 \end{smallmatrix}\right)$?

Motivation The first question is a convex analytic proof to the (known fact) that $P_{2,4}=\Sigma_{2,4}$, while the second question is to prove $P_{2,6}=\Sigma_{2,6}$. Below, I describe the origin of the problem for the former.

\end{matrix}\right)$end{smallmatrix}\right)$. The above proof of which I'm seeking a convex analytic proof is equivalent to the assertion $P_{2,4}=\Sigma_{2,4}$.
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Elementary Convex Analytic or linear algebraic proof that a certain psd matrix is a sum of rank 1 psd matrices

Do

Can you know of prove the following using techniques from convex analysis or linear algebra? I was originally seeking an elementary proofthat a positive semidefinite real symmetric , but I think it is better to broaden the scope for this bounty question.

A psd matrix of the form $\left(\begin{smallmatrix} \end{smallmatrix}\right)$ can be written as the sum of finitely many rank 1 matrices of the form $\left(\begin{matrix}

I came upon this problem as a convex analytic approach to the (known fact) that $P_{2,4}=\Sigma_{2,4}$. Here $P_{2,4}$ is the cone of nonnegative-valued binary quartics. $\Sigma_{2,4}$ is the cone of binary quartics that are sums of squares. (A binary quartic is a homogeneous polynomial in 2 variables of degree 4). Obviously $P_{2,4}\subseteq \Sigma_{2,4}$. The standard proof that equality holds is by dehomonogizing and applying the Fundamental theorem of algebra.

Since the cone $\Sigma_{2,4}$ is closed in the vector space ${\mathbb{R}}[x,y]_4$ of homogeneous polynomials in 2 variables of degree 4, a separation theorem in convex geometry provides a necessary and sufficient condition for a binary quartic $f$ to lie in $\Sigma_{2,4}$. This condition is that, for any linear functional $T$ which spans an extremal ray of the dual cone $\Sigma_{2,4}^{\vee}$, we have $T(f)\ge 0$.

The cone of positive semidefinite matrices of the form $\left(\begin{smallmatrix}\end{smallmatrix}\right)$ is dual isomorphic to the cone of sums of squares which are homoegneous polynomials $\Sigma_{2,4}^{\vee}$. Under this isomorphism, point evaluations correspond to rank 1 matrices of degree 4 in two real variablesthe form $\left(\begin{matrix}x^4 & x^3 y & x^2 y^2\\x^3 y & x^2 y^2 & x y^3 \\x^2 y^2 & x y^3 & y^4\end{matrix}\right)$. The above result proof of which I am I'm seeking an elementary a convex analytic proof will imply (the known fact) that every nonnegative-valued polynomial in degree 4 and two variables is a sum of squares.equivalent to the assertion $P_{2,4}=\Sigma_{2,4}$.
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Do you know of an elementary proof that a positive semidefinite real symmetric matrix of the form $\left(\begin{matrix} \left(\begin{smallmatrix} a & b & cc\\ b & c & d \\ c & d & e \end{matrix}\right)$ end{smallmatrix}\right)$ can be written as the sum of finitely many matrices of the form $\left(\begin{matrix} x^4 & x^3 y & x^2 y^2y^2\\ x^3 y & x^2 y^2 & x y^3 \\ x^2 y^2 & x y^3 & y^4 \end{matrix}\right)$?

I came upon this question as end{matrix}\right)$?

The cone of positive semidefinite matrices of the form $\left(\begin{smallmatrix} a & b & c\\ b & c & d \\ c & d & e \end{smallmatrix}\right)$ is dual to the cone of sums of squares which are homoegneous polynomials of degree 4 in two real variables. The above result of which I am seeking an elementary proof will imply (the known fact) that every nonnegative-valued real homogeneous polynomial in 2 variables and of degree 4 and two variables is a sum of squares.
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