lec5 A=LUA=LUA=LU $\forall invertibleinvertibleinvertible A ,,, \exists PA=LU$ transpose (A⊤)ij=Aji\left(A^{\top}\right)_{i j}=A j i (A⊤)ij​=Aji Symmetrix Matrix A⊤=AA^{\top}=A A⊤=A R⊤RR^{\top}RR⊤R is

矩阵乘法的性质 顺序不能改变，可以任意添加括号 (AB)C=A(BC)(AB)C=A(BC) (AB)C=A(BC) 矩阵乘法的定义 从单个元素 Cij=∑kAikBkjC_{ij}=\sum_{k}A_{ik}B_{kj} Cij​=k∑​Aik​Bkj​ 从列 A * col 1 of B = col 1 of C cols of B 对 cols of A的线性组合 从行 row 1 of

lec27 complex vector whenever we transpose, we also take conjugates: zˉ⊤z\bar{z}^\top zzˉ⊤z , because z1ˉz1=∣z1∣2\bar{z_1}z_1=|z_1|^2z1​ˉ​z1​=∣z1​∣2 . $\bar{z}^\top z denoteasdenote asdenoteas z^Hz$

lec26 Eigenvector of symmetric matrices A symmetric matrix has only real eigenvalues The eigenvectors can be chosen orthonormal proof Orthogonality of Eigenvectors of a Symmetric Matrix Correspo

lec4 基础性质 1 ⁍⁍ ⁍ proof. (AB)(B−1A−1)=A(BB−1)A−1=AA−1=I(A B)\left(B^{-1} A^{-1}\right)=A\left(B B^{-1}\right) A^{-1}=A A^{-1}=I (AB)(B−1A−1)=A(BB−1)A−1=AA−1=I 3 (AB)⊤=B⊤A⊤(A B)^{\top}=B^{\top} A^{\top}

lec24 Markov Matrix A=[.1.01.3.2.99.3.70.4]A=\left[\begin{array}{ccc}.1 & .01 & .3 \\.2 & .99 & .3 \\.7 & 0 & .4\end{array}\right] A=​.1.2.7​.01.990​.3.3.4​​ properties All en

lec9 Independent no combination give a zero vector (expect 000 ). If columns of AAA are independent, N(A)=0N(A) = {0}N(A)=0 . Span vectors span a space means the space consists of all combinations of

模板题 #include <bits/stdc++.h> #define rep(i, a, b) for (int i = (a); i <= (b); i++) #define per(i, a, b) for (int i = (a); i >= (b); i--) #define D(x) cout <<

转换flac 无损转换+保留封面 for file in ./*.flac do ffmpeg -i " $file" -c:v copy -c:a alac "$ {file%.*}.m4a" done

求解 $ ax≡b(modp)a^x \equiv b \pmod pax≡b(modp) 考虑分块，令 考虑分块，令 考虑分块，令 x=A⌈p⌉−Bx = A\lceil\sqrt{p}\rceil - Bx=A⌈p​⌉−B $ 其中 0≤B<⌈p⌉0 \le B < \lceil\sqrt{p}\rceil0≤B<⌈p​⌉ 那么 $ aA⌈p⌉−B≡b(modp)a ^ {A