利用排列选取构造矩阵

青青子衿

1. MatrixForm[{#1[[1]], #2[[2]], #3[[3]]}] & @@@
2. Permutations[{({{x^2, x*y, x*z}, {x*y, y^2, y*z}, {x*z, y*z,
3.       z^2}}), ({{a, h, g}, {h, b, f}, {g, f, c}}), ({{a1, h1,
4.       g1}, {h1, b1, f1}, {g1, f1, c1}})}]
5. MatrixForm[
6.    MapIndexed[{({{x^2, x*y, x*z}, {x*y, y^2, y*z}, {x*z, y*z,
7.           z^2}}), ({{a, h, g}, {h, b, f}, {g, f, c}}), ({{a1, h1,
8.           g1}, {h1, b1, f1}, {g1, f1, c1}})}[[#1, #2[[1]]]] &, #]] & /@
9.   Permutations[{1, 2, 3}]

复制代码

\begin{gather*}
\boldsymbol{P}=\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}^{\mathrm{T}}\\
\quad\boldsymbol{Q}_0=\begin{pmatrix}
A & H & G \\
H & B & F \\
G & F & C \\
\end{pmatrix}
\qquad
\boldsymbol{Q}_1=\begin{pmatrix}
A_1 & H_1 & G_1 \\
H_1 & B_1 & F_1 \\
G_1 & F_1 & C_1 \\
\end{pmatrix}
\\
\end{gather*}

\begin{align*}
I(\boldsymbol{P},\boldsymbol{Q}_0,\boldsymbol{Q}_1)&=\begin{vmatrix}
x^2 & x y & x z \\
H_1 & B_1 & F_1 \\
G & F & C \\
\end{vmatrix}+\begin{vmatrix}
x^2 & x y & x z \\
H & B & F \\
G_1 & F_1 & C_1 \\
\end{vmatrix}\\
\,
&\qquad+
\begin{vmatrix}
A_1 & H_1 & G_1 \\
x y & y^2 & y z \\
G & F & C \\
\end{vmatrix}+
\begin{vmatrix}
A & H & G \\
x y & y^2 & y z \\
G_1 & F_1 & C_1 \\
\end{vmatrix}\\
\,
&\qquad\quad+
\begin{vmatrix}
A_1 & H_1 & G_1 \\
H & B & F \\
x z & y z & z^2 \\
\end{vmatrix}
+\begin{vmatrix}
A & H & G \\
H_1 & B_1 & F_1 \\
x z & y z & z^2 \\
\end{vmatrix}\\
\end{align*}