NCMatrixOfQuadratic[ ${\cal {Q}}$, $\{H_1,\ldots,H_n \}$ ]

Aliases: None.

Description: NCMatrixOfQuadratic[ ${\mathcal{Q}}$,{ $H_1,H_2,\ldots,H_n$} ] gives a vector matrix factorization of a symmetric quadratic function $\mathcal{Q}$ in noncommutative variables $\vec{H} = \{ H_1,H_2,\ldots,H_n \}$ and their transposes.
NCMatrixOfQuadratic[ $\mathcal{Q}$, $\{H_1,H_2,\ldots,H_n \}$ ], generates the list {left border vector, coefficient matrix, right border vector}. That is, $Q$ is factored into the vector-matrix-vector product $V[\vec{H}]^T M_\mathcal{Q} V[\vec{H}]$. The vector $V[\vec{H}]$ is linear in $\vec{H}$ and is called a border vector of the quadratic function $\mathcal{Q}$. The matrix $M_\mathcal{Q}$ is called the coefficient matrix of the quadratic function $\mathcal{Q}$.

Arguments: Each term of $\mathcal{Q}$ is assumed to be a quadratic expression in terms of the variables $H_1,H_2,\ldots,H_n$ and their transposes ( $\mathcal{Q}$ is homogeneous).
For example, suppose that $\mathcal{Q}=3 tp[x]**y+3 tp[y]**x$ and that
$\vec{H} = \{x,y\}$. Then, NCMatrixOfQuadratic[ $\mathcal{Q}$, $\vec{H}$ ] gives

$$\{\{\{tp[x], tp[y]\}\},\{\{0,3\},\{3,0\}\},\{\{x\},\{y\} \} \}.$$

In MatrixForm, this looks like

$$\left( tp[x] \;\;\; tp[y]\right) * \left( \begin{array}{cc}0 & 3 \ 3 & 0\end{array} \right)* \left( \begin{array}{c}x \ y\end{array}\right).$$

In general, suppose $\mathcal{Q}$ is a quadratic function of two variables, $\vec{H} = \{H,K\}$, with all transpose elements ${H}^T,\;{K}^T$ occuring before all non-transpose elements. Then NCMatrixOfQuadratic will return the left border vector $V[\vec{H}]^T$, the matrix $M_\mathcal{Q}$, and the right vector $V[\vec{H}]$ where

$$M_\mathcal{Q} := \left( \begin{array}{ccccccc} A_{11}&A_{12}&\cdo... ...\cdots& A_{\ell_1,n}^T& A_{\ell_1+1,n}^T & \cdots & A_{n,n} \end{array} \right)$$

$$\textrm{ and } V[\vec{H}]:= \left( \begin{array}{c} HL^1_1\ HL^1_2\ \cdots\ HL^1_{\ell_1}\ KL^2_1\ \cdots \ KL^2_{\ell_2} \end{array}\right)$$

for some $L_i^j , i=1,\dots,\ell_j$. The $L_i^j , i=1,\dots,\ell_j$ are called the coefficients of the border vector. The $L_i^1$ corresponding to $H$ are distinct and only one may be the identity matrix (equivalently for the $L_i^2$ corresponding to $K$). The border vector $V$ is the vector composed of $H$, $K$ and $L_i^j$. The matrix $M_\mathcal{Q}$ is the matrix with $A_{i,j}$ entries.
Noncommutative quadratics which are not hereditary have a similar representation (which takes more space to write) for such a quadratic in $H,K$. For example, the border vector for a quadratic in $H$, $H^T$, $K$, $K^T$ has the form

$$V[H,K] = \begin{matrix}V_1 & V_2 \end{matrix}$$

where we have

$$V_1= \big ( (L^1_1)^T H^T,\cdots,(L^1_{\ell_1})^T H^T,(L^2_1)^T K^T,\cdots, L(^2_{\ell_2})^TK^T \big )$$

and

$$V_2 = \big ( \tilde{L}^1_1 H ,\cdots,\tilde{L}^1_{\ell_1} H,\tilde{L}^2_1 K,\cdots,\tilde{L}^2_{\ell_2} K \big ).$$

We should emphasize that the size of the $M_\mathcal{Q}$ representation of a noncommutative quadratic functions $\mathcal{Q}[H_1, \dots, H_k]$ depends on the particular quadratic and not only on the number of arguments of the quadratic. There are noncommutative quadratic functions in one variable which have a representation with $M_{\mathcal{Q}}$ a 102 $\times$ 102 matrix.
The basic idea of NCMatrixOfQuadratic is that it searches for terms of form

$$Left**X**Middle**Y**Right$$

where $X = H_i$ or $H_i^T$ and $Y = H_j$ or $H_j^T$ for $1 \le (i,j) \le n$. Terms of the form $Left**X$ and $Y**Right$ are used to form the left and right vectors. Each time the search finds a unique $Right$ ($Left$) term causes the length of the right (left) border vector to be increased by one. The term $Middle$ becomes the entries in the matrix $M_{\mathcal{Q}}$.

Comments / Limitations: NCMatrixOfQuadratic will try to symmetrize the resulting matrix $M_{\mathcal{Q}}$. If NCMatrixOfQuadratic is unable to do this, an error message will be printed and $\{$ leftvector, matrix, rightvector $\}$ will be returned, where matrix is not symmetric and leftvector is not necessarily the transpose of rightvector. The vector-matrix-vector product should still be equal to the orginal quadratic expression.