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Is there a general rule for how to write high order polynomials in matrix form?

For example a linear combination of parameters:

$$a_1 x_1+a_2 x_2+a_3 x_3 + \cdots+ a_n x_n$$

Can be written as

$$\sum^n_{i=1} a_i x_i = \vec{a}^T\vec{x} $$

Second order forms are given by

$$ (a_1 x_1+a_2 x_2+a_3 x_3 + \cdots+ a_n x_n)^2 = \vec{x}^T {\mathbf A} \vec{x}$$

Which ensures all combinations of second order terms. What about the higher orders? i.e.

$$(a_1 x_1+a_2 x_2+a_3 x_3 + \dots +a_n x_n)^k$$

What forms ensure all combinations of terms. Is there a general rule to this? Does it have a name?

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  • $\begingroup$ You might be interested in Einstein notation. $\endgroup$ – Luca Bressan Sep 15 '16 at 14:51
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    $\begingroup$ Notice that $(\vec{a}\otimes\ldots\otimes \vec{a})^t(\vec{x}\otimes\ldots\otimes \vec{x})=(\vec{a}^t\vec{x})^k$, where $\otimes$ means the kronecker product of the column matrices $\vec{a}_{n\times 1},\vec{x}_{n\times 1}$. $\endgroup$ – Daniel Sep 15 '16 at 16:05
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Linear Form

$$ \begin{pmatrix} a & b \\ \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}= ax+by$$

Quadratic Form

$$ \begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} a & b \\ b & c \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}= ax^2+2bxy+cy^2$$

Cubic Form

$$ \begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} ax+by & bx+cy \\ bx+cy & cx+dy \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}= ax^3+3bx^2y+3cxy^2+dy^3$$

Quartic Form

$$ \begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} ax^2+2bxy+cy^2 & bx^2+2cxy+dy^2 \\ bx^2+2cxy+dy^2 & cx^2+2dxy+ey^2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$

$m$-tuple Form

$$ \binom{m}{i_{1},i_{2}, \ldots , i_{n}} a_{i_{1} i_{2} \ldots i_{n}} x_{1}^{i_{1}} x_{2}^{i_{2}} \ldots x_{n}^{i_{n}}$$

where $\boldsymbol{x} \in \mathbb{R}^{n}$ and $i_{1}+i_{2}+\ldots+i_{n}=m$

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I like what has been said already by @Daniel and @Ng Chung Tak but let me rephrase their answers.

We'll consider tensor products of the 1-form $a=(a_1,\ldots a_n)\in V^*$. A bilinear form $b=a\otimes a$ (which we'll call a "second order form" when supplied with the same $x\in V$ in both argument slots) can indeed be represented by a matrix equal to the outer product of the coordinates of $a$. A multilinear form $m=\otimes^ka$ would be represented by a multidimensional array.

Alternatively, you can look at this as an application of Multinomial theorem and multi-index notation

$$ (a_1x_1+\cdots+a_nx_n)^k = \sum_{|\alpha|=k}{k \choose \alpha}a^\alpha x^\alpha $$

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