On page 212:
- Prove that every nonnegative bilinear form of rank $r$ in the space $\mathbf R_n$ can be represented as a sum of $r$ nonnegative bilinear forms of rank $1$.
According to the hint given at the end of the book, I can prove the statement when leaving out the condition "nonnegative", as follows.
$$A(\pmb x, \pmb y) = \sum_{i,j = 1}^n a_{ij} x_i y_j = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \begin{bmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_n \end{bmatrix} \\ = \pmb x^T A \pmb y = \pmb x^T \sum_{k = 1}^r L^{(k)}(\pmb y) \pmb e^{(k)} \\ = L^{(1)}(\pmb y) \pmb x^T \pmb e^{(1)} + L^{(2)}(\pmb y) \pmb x^T \pmb e^{(2)} + \cdots + L^{(r)}(\pmb y) \pmb x^T \pmb e^{(r)}, $$
where $\pmb e^{(1)}, \pmb e^{(2)}, \cdots, \pmb e^{(r)}$ is a basis for the range of $A$, and $L^{(1)}(\pmb y), L^{(2)}(\pmb y), \cdots, L^{(r)}(\pmb y)$ is a series of numeral functions of one vector argument. It can be proved that every $L^{(k)}(\pmb y)$ is a linear form. Thus, each term in the summation $$L^{(k)}(\pmb y) \pmb x^T \pmb e^{(k)} = \sum_{j = 1}^n \ell_j^{(k)} y_j \sum_{i = 1}^n e_i^{(k)} x_i = \sum_{i, j = 1}^n e_i^{(k)} \ell_j^{(k)} x_i y_j. $$
And it's easy to verify that the following matrix is of rank $1$, as required. $$\begin{bmatrix} e_1^{(k)} \ell_1^{(k)} & e_1^{(k)} \ell_2^{(k)} & \cdots & e_1^{(k)} \ell_n^{(k)} \\ e_2^{(k)} \ell_1^{(k)} & e_2^{(k)} \ell_2^{(k)} & \cdots & e_2^{(k)} \ell_n^{(k)} \\ \vdots & \vdots & \ddots & \vdots \\ e_n^{(k)} \ell_1^{(k)} & e_n^{(k)} \ell_2^{(k)} & \cdots & e_n^{(k)} \ell_n^{(k)} \end{bmatrix}$$
But I have no idea how to deal with the condition "nonnegative". Any ideas? Thanks.