第2不変量
$ I_2^{\bm A}:=\frac12\left(\left({\mathrm{tr}\bm A}\right)^2-\mathrm{tr}\left(\bm A^2\right)\right)
性質
3次元のとき$ I_2^{\bm A}=\lambda_0^{\bm A}\lambda_1^{\bm A}+\lambda_1^{\bm A}\lambda_2^{\bm A}+\lambda_2^{\bm A}\lambda_0^{\bm A}
任意基底で成分表示すると
$ I_2^{\bm A}=\frac12\left({{A_i}^i}^2-{A_i}^j{A_j}^i\right)
2次元の時
$ =\frac12\left(2{A_1}^1{A_2}^2-{A_1}^2{A_2}^1-{A_2}^1{A_1}^2\right)
$ ={A_1}^1{A_2}^2-{A_1}^2{A_2}^1
$ =I_3^{\bm A}
$ =\det\bm A
3次元の時
$ =\frac12\left(2{A_1}^1{A_2}^2+2{A_1}^1{A_3}^3+2{A_3}^3{A_1}^1-{A_1}^2{A_2}^1-{A_2}^3{A_2}^3-{A_3}^1{A_3}^1-{A_1}^3{A_1}^3-{A_3}^2{A_3}^2-{A_2}^1{A_2}^1\right)