Skip to content

Fix the expression of Q-lambda*I in CH21 #21

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
wants to merge 1 commit into
base: master
Choose a base branch
from
Open

Fix the expression of Q-lambda*I in CH21 #21

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
2 changes: 1 addition & 1 deletion chapter21.ipynb
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -62,7 +62,7 @@
" \n",
" 对于$Q$矩阵，有$\\begin{cases}\\lambda_1+\\lambda_2&=0\\\\\\lambda_1\\cdot\\lambda_2&=1\\end{cases}$，再来思考特征值与特征向量的由来，哪些向量旋转$90^\\circ$后与自己平行，于是遇到了麻烦，并没有这种向量，也没有这样的特征值来满足前面的方程组。\n",
" \n",
" 我们来按部就班的计算，$\\det(Q-\\lambda I)=\\begin{vmatrix}\\lambda&-1\\\\1&\\lambda\\end{vmatrix}=\\lambda^2+1=0$，于是特征值为$\\lambda_1=i, \\lambda_2=-i$，我们看到这两个值满足迹与行列式的方程组，即使矩阵全是实数，其特征值也可能不是实数。本例中即出现了一对共轭负数，我们可以说，如果矩阵越接近对称，那么特征值就是实数。如果矩阵越不对称，就像本例，$Q^T=-Q$，这是一个反对称的矩阵，于是我得到了纯虚的特征值，这是极端情况，通常我们见到的矩阵是介于对称与反对称之间的。\n",
" 我们来按部就班的计算，$\\det(Q-\\lambda I)=\\begin{vmatrix}-\\lambda&-1\\\\1&-\\lambda\\end{vmatrix}=\\lambda^2+1=0$，于是特征值为$\\lambda_1=i, \\lambda_2=-i$，我们看到这两个值满足迹与行列式的方程组，即使矩阵全是实数，其特征值也可能不是实数。本例中即出现了一对共轭负数，我们可以说，如果矩阵越接近对称，那么特征值就是实数。如果矩阵越不对称，就像本例，$Q^T=-Q$，这是一个反对称的矩阵，于是我得到了纯虚的特征值，这是极端情况，通常我们见到的矩阵是介于对称与反对称之间的。\n",
" \n",
" 于是我们看到，对于好的矩阵（置换矩阵）有实特征值及正交的特征向量，对于不好的矩阵（$90^\\circ$旋转矩阵）有纯虚的特征值。\n",
" \n",
Expand Down