幂法求矩阵主特征值及特征向量 - 详细步骤与案例

首先任取一个非零向量作为初始向量$x^{(0)}$，这里取$x^{(0)}=[1,1,1]^T$。/n/n第一次迭代：/n$$/ny^{(1)}=Ax^{(0)}=/begin{bmatrix}/n7 & 3 & -2 ///n3 & 4 & -1 ///n-2 & -1 & 3 ///n/end{bmatrix}/begin{bmatrix}/n1 ///n1 ///n1 ///n/end{bmatrix}=/begin{bmatrix}/n8 ///n6 ///n0 ///n/end{bmatrix}/n$$/n$$/n//lambda^{(1)}=y^{(1)}_1/ x^{(0)}_1=8/1=8/n$$/n$$/nx^{(1)} = y^{(1)}///left//Vert y^{(1)}//right//Vert =/begin{bmatrix}/n8/10 ///n6/10 ///n0 ///n/end{bmatrix}=/begin{bmatrix}/n0.8 ///n0.6 ///n0 ///n/end{bmatrix}/n$$/n/n第二次迭代：/n$$/ny^{(2)}=Ax^{(1)}=/begin{bmatrix}/n7 & 3 & -2 ///n3 & 4 & -1 ///n-2 & -1 & 3 ///n/end{bmatrix}/begin{bmatrix}/n0.8 ///n0.6 ///n0 ///n/end{bmatrix}=/begin{bmatrix}/n6.8 ///n5.4 ///n-1.2 ///n/end{bmatrix}/n$$/n$$/n//lambda^{(2)}=y^{(2)}_1/ x^{(1)}_1=6.8/0.8=8.5/n$$/n$$/nx^{(2)} = y^{(2)}///left//Vert y^{(2)}//right//Vert =/begin{bmatrix}/n6.8/8.7 ///n5.4/8.7 ///n-1.2/8.7 ///n/end{bmatrix}=/begin{bmatrix}/n0.7816092 ///n0.6206897 ///n-0.0896552 ///n/end{bmatrix}/n$$/n/n第三次迭代：/n$$/ny^{(3)}=Ax^{(2)}=/begin{bmatrix}/n7 & 3 & -2 ///n3 & 4 & -1 ///n-2 & -1 & 3 ///n/end{bmatrix}/begin{bmatrix}/n0.7816092 ///n0.6206897 ///n-0.0896552 ///n/end{bmatrix}=/begin{bmatrix}/n6.9586207 ///n5.5114943 ///n-1.0143678 ///n/end{bmatrix}/n$$/n$$/n//lambda^{(3)}=y^{(3)}_1/ x^{(2)}_1=6.9586207/0.7816092=8.898/n$$/n$$/nx^{(3)} = y^{(3)}///left//Vert y^{(3)}//right//Vert =/begin{bmatrix}/n6.9586207/8.836 ///n5.5114943/8.836 ///n-1.0143678/8.836 ///n/end{bmatrix}=/begin{bmatrix}/n0.7862574 ///n0.6210274 ///n-0.0087686 ///n/end{bmatrix}/n$$/n/n经过三次迭代，特征值已经稳定到小数点后三位，因此得到主特征值为8.898，对应的特征向量为$/begin{bmatrix}/n0.7862574 ///n0.6210274 ///n-0.0087686 ///n/end{bmatrix}$。