ベクトル・行列の計算

 

定義した変数があるかないかanamesを使い、あればdeleteします。

anames(All,User);

{}

$を使ってベクトルを定義します。

A:=matrix([a[i] $ i=1..3]);
B:=matrix([b[i] $ i=1..3]);

matrix([[a[1]], [a[2]], [a[3]]])
matrix([[b[1]], [b[2]], [b[3]]])

transposeを使って転置します。

A:=transpose(A);
B:=transpose(B);

matrix([[a[1], a[2], a[3]]])
matrix([[b[1], b[2], b[3]]])

linalg::scalarProductを使って内積を求めます。

linalg::scalarProduct(A,B);

conjugate(b[1])*a[1] + conjugate(b[2])*a[2] + conjugate(b[3])*a[3]

実数計算に限定します。

linalg::scalarProduct(A,B,Real);

a[1]*b[1] + a[2]*b[2] + a[3]*b[3]

linalg::crossProductを使って外積を求めます。

linalg::crossProduct(A,B);

matrix([[a[2]*b[3] - a[3]*b[2], a[3]*b[1] - a[1]*b[3], a[1]*b[2] - a[2]*b[1]]])

L2ノルムを求めます。

norm(A,2);

(abs(a[1])^2 + abs(a[2])^2 + abs(a[3])^2)^(1/2)

$を使って行列を定義します。

delete A,B;
A:=matrix([[a[j,i] $ j=1..3] $ i=1..3]);

matrix([[a[1, 1], a[2, 1], a[3, 1]], [a[1, 2], a[2, 2], a[3, 2]], [a[1, 3], a[2, 3], a[3, 3]]])

行列式を求めます。

det(A);

a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]

 

inverse用いて逆行列を求めます。

B:=inverse(A);// A^(-1)か1/AでもOK

matrix([[(a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), -(a[2, 1]*a[3, 3] - a[3, 1]*a[2, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), (a[2, 1]*a[3, 2] - a[2, 2]*a[3, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1])], [-(a[1, 2]*a[3, 3] - a[1, 3]*a[3, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), (a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), -(a[1, 1]*a[3, 2] - a[1, 2]*a[3, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1])], [(a[1, 2]*a[2, 3] - a[1, 3]*a[2, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), -(a[1, 1]*a[2, 3] - a[2, 1]*a[1, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), (a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1])]])

行列の掛け算で検算します。

A*B

matrix([[((a[1, 2]*a[2, 3] - a[1, 3]*a[2, 2])*a[3, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[1, 2]*a[3, 3] - a[1, 3]*a[3, 2])*a[2, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) + ((a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2])*a[1, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), ((a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1])*a[2, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[1, 1]*a[2, 3] - a[2, 1]*a[1, 3])*a[3, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[2, 1]*a[3, 3] - a[3, 1]*a[2, 3])*a[1, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), ((a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1])*a[3, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[1, 1]*a[3, 2] - a[1, 2]*a[3, 1])*a[2, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) + ((a[2, 1]*a[3, 2] - a[2, 2]*a[3, 1])*a[1, 1])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1])], [((a[1, 2]*a[2, 3] - a[1, 3]*a[2, 2])*a[3, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[1, 2]*a[3, 3] - a[1, 3]*a[3, 2])*a[2, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) + ((a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2])*a[1, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), ((a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1])*a[2, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[1, 1]*a[2, 3] - a[2, 1]*a[1, 3])*a[3, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[2, 1]*a[3, 3] - a[3, 1]*a[2, 3])*a[1, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), ((a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1])*a[3, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[1, 1]*a[3, 2] - a[1, 2]*a[3, 1])*a[2, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) + ((a[2, 1]*a[3, 2] - a[2, 2]*a[3, 1])*a[1, 2])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1])], [((a[1, 2]*a[2, 3] - a[1, 3]*a[2, 2])*a[3, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[1, 2]*a[3, 3] - a[1, 3]*a[3, 2])*a[2, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) + ((a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2])*a[1, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), ((a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1])*a[2, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[1, 1]*a[2, 3] - a[2, 1]*a[1, 3])*a[3, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[2, 1]*a[3, 3] - a[3, 1]*a[2, 3])*a[1, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]), ((a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1])*a[3, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) - ((a[1, 1]*a[3, 2] - a[1, 2]*a[3, 1])*a[2, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1]) + ((a[2, 1]*a[3, 2] - a[2, 2]*a[3, 1])*a[1, 3])/(a[1, 1]*a[2, 2]*a[3, 3] - a[1, 1]*a[2, 3]*a[3, 2] - a[1, 2]*a[2, 1]*a[3, 3] + a[1, 2]*a[3, 1]*a[2, 3] + a[2, 1]*a[1, 3]*a[3, 2] - a[1, 3]*a[2, 2]*a[3, 1])]])

simplifyを使って簡単にします。

simplify(A*B);

matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])

Frobeniusのノルムを求めます。

norm(A,Frobenius);

(abs(a[1, 1])^2 + abs(a[1, 2])^2 + abs(a[2, 1])^2 + abs(a[1, 3])^2 + abs(a[2, 2])^2 + abs(a[3, 1])^2 + abs(a[2, 3])^2 + abs(a[3, 2])^2 + abs(a[3, 3])^2)^(1/2)

linalg::eigenvaluesを使って固有値を求めます。

linalg::eigenvalues(A);

{((a[1, 1] + a[2, 2] + a[3, 3])^3/27 + (((a[1, 1] + a[2, 2] + a[3, 3])^3/27 - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^2 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)^3)^(1/2) - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^(1/3) + ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)/((a[1, 1] + a[2, 2] + a[3, 3])^3/27 + (((a[1, 1] + a[2, 2] + a[3, 3])^3/27 - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^2 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)^3)^(1/2) - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^(1/3) + a[1, 1]/3 + a[2, 2]/3 + a[3, 3]/3, a[1, 1]/3 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)/(2*((a[1, 1] + a[2, 2] + a[3, 3])^3/27 + (((a[1, 1] + a[2, 2] + a[3, 3])^3/27 - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^2 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)^3)^(1/2) - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^(1/3)) - ((a[1, 1] + a[2, 2] + a[3, 3])^3/27 + (((a[1, 1] + a[2, 2] + a[3, 3])^3/27 - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^2 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)^3)^(1/2) - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^(1/3)/2 + a[2, 2]/3 + a[3, 3]/3 - (3^(1/2)*(((a[1, 1] + a[2, 2] + a[3, 3])^3/27 + (((a[1, 1] + a[2, 2] + a[3, 3])^3/27 - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^2 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)^3)^(1/2) - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^(1/3) - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)/((a[1, 1] + a[2, 2] + a[3, 3])^3/27 + (((a[1, 1] + a[2, 2] + a[3, 3])^3/27 - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^2 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)^3)^(1/2) - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^(1/3))*I)/2, a[1, 1]/3 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)/(2*((a[1, 1] + a[2, 2] + a[3, 3])^3/27 + (((a[1, 1] + a[2, 2] + a[3, 3])^3/27 - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^2 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)^3)^(1/2) - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^(1/3)) - ((a[1, 1] + a[2, 2] + a[3, 3])^3/27 + (((a[1, 1] + a[2, 2] + a[3, 3])^3/27 - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^2 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)^3)^(1/2) - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^(1/3)/2 + a[2, 2]/3 + a[3, 3]/3 + (3^(1/2)*(((a[1, 1] + a[2, 2] + a[3, 3])^3/27 + (((a[1, 1] + a[2, 2] + a[3, 3])^3/27 - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^2 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)^3)^(1/2) - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^(1/3) - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)/((a[1, 1] + a[2, 2] + a[3, 3])^3/27 + (((a[1, 1] + a[2, 2] + a[3, 3])^3/27 - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^2 - ((a[1, 1] + a[2, 2] + a[3, 3])^2/9 - (a[1, 1]*a[2, 2])/3 + (a[1, 2]*a[2, 1])/3 - (a[1, 1]*a[3, 3])/3 + (a[1, 3]*a[3, 1])/3 - (a[2, 2]*a[3, 3])/3 + (a[2, 3]*a[3, 2])/3)^3)^(1/2) - ((a[1, 1] + a[2, 2] + a[3, 3])*(a[1, 1]*a[2, 2] - a[1, 2]*a[2, 1] + a[1, 1]*a[3, 3] - a[1, 3]*a[3, 1] + a[2, 2]*a[3, 3] - a[2, 3]*a[3, 2]))/6 + (a[1, 1]*a[2, 2]*a[3, 3])/2 - (a[1, 1]*a[2, 3]*a[3, 2])/2 - (a[1, 2]*a[2, 1]*a[3, 3])/2 + (a[1, 2]*a[3, 1]*a[2, 3])/2 + (a[2, 1]*a[1, 3]*a[3, 2])/2 - (a[1, 3]*a[2, 2]*a[3, 1])/2)^(1/3))*I)/2}