行列と双線形写像 2次形式対応する行列

学習環境

ラング線形代数学(下) (ちくま学芸文庫) (S.ラング(著)、芹沢正三(翻訳)、筑摩書房)の8章(行列と双線形写像)、2(2次形式)、練習問題2の解答を求めてみる。

\begin{array}{l} [\begin{matrix} x & y & z \end{matrix}] [\begin{matrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = x^{2} - 3 x y + 4 y^{2} \\ a_{i j} = a_{j i} (i \neq j) \end{array}

[\begin{matrix} c_{11} x + c_{21} y + c_{31} z & c_{12} x + c_{22} y + c_{32} z & c_{13} x + c_{23} y + c_{33} z \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = x^{2} - 3 x y + 4 y^{2}

\begin{array}{l} c_{11} x^{2} + (c_{21} + c_{12}) x y + (c_{31} + c_{13}) x z + c_{22} y^{2} + (c_{32} + c_{23}) y z + c_{33} z^{2} \\ = x^{2} - 3 x y + 4 y^{2} \end{array}

\begin{array}{l} c_{11} = 1 \\ c_{21} + c_{12} = - 3 \\ c_{31} + c_{13} = 0 \\ c_{22} = 4 \\ c_{32} + c_{23} = 0 \\ c_{33} = 0 \end{array}

よって、 2次形式

f (X) = x^{2} - 3 x y + 4 y^{2}

に対応する行列は、

C = [\begin{matrix} 1 & - \frac{3}{2} & 0 \\ - \frac{3}{2} & 4 & 0 \\ 0 & 0 & 0 \end{matrix}]

コード(Wolfram Language, Jupyter)

xv = {x, y, z}

{x, y, z}

c = {{c11, c12, c13}, {c12, c22, c23}, {c13, c23, c33}}

{{c11, c12, c13}, {c12, c22, c23}, {c13, c23, c33}}

xv . c . xv

Simplify[%]

Solve[{c11 == 1, 2c12 == -3, c22 == 4, 2c13 == 0, 2c23 == 0, c33 == 0}, {c11, c12, c13, c22, c23, c33}]

MatrixForm[c /. %]