HackMD - Collaborative Markdown Knowledge Base

## Partial Fraction decomposition $$ \frac{12x}{(x-1)(x-2)(x-3)}=\frac A{x-1}+\frac B{x-2}+\frac C{x-3} $$ $A=6$ $B=-24$ $C=18$ ## Lagurange Interprolation Polynomial. Find a linear function $f(x)=\frac{x-a}b$ such that $$ f(5)=1; f(7)=0 $$ $$ f(x)=\frac{x-7}{5-7} $$ Another exercise: Find a quadratic polynomial such that $$ f(5)=1; f(7)=0; f(9)=0 $$ $$ f(x)=\frac{(x-7)(x-9)}{(5-7)(5-9)} $$ Another exercise: Find a quadratic polynomial $f(x)$ such that $$ f(5)=2, f(7)=3, f(9)=4 $$ ? |$f$|$f_1$|$f_2$|$f_3$|$(x-1)/2$ -|-|-|-|-|- x=5|2|2|0|0|2 x=7|3|0|3|0|3 x=9|4|0|0|4|4 $$ f_1(x)=2\cdot\frac{(x-7)(x-9)}{(5-7)(5-9)} $$ $$ f_2(x)=3\cdot\frac{(x-5)(x-9)}{(7-5)(7-9)} $$ $$ f_3(x)=4\cdot\frac{(x-5)(x-7)}{(9-5)(9-7)} $$ $$ f(x)=2\cdot\frac{(x-7)(x-9)}{(5-7)(5-9)}+3\cdot\frac{(x-5)(x-9)}{(7-5)(7-9)}+4\cdot\frac{(x-5)(x-7)}{(9-5)(9-7)} $$ 这个多项式是不是 $f(x)$ 的唯一构造？为了说明**唯一性**，我们不妨假设有两个二次多项式 $f,g$ 使得 ? |$f$|$g$|$f-g$ -|-|-|- x=5|2|2|0 x=7|3|3|0 x=9|4|4|0 因为 $f$ 与$g$ 都是至多二次的多项式，那$f-g$也是最多二次的多项式 $$ f(x)-g(x)=ax^2+bx+c $$ ? |$ax^2+bx+c$|$(x-5)(x-7)$|$(ax^2+bx+c)-a(x-5)(x-7)$ -|-|-|- x=5|0|0|0 x=7|0|0|0 x=9|0|$\neq 0$|0 ## Diagonalization #### 一个拉格朗日插值多项式的小性质把 $t$当成常数; 拉格朗日插值多项式：假设 $$ \lambda_1,\lambda_2,\cdots \lambda_n $$ 是n个不同的数字，找到一个$n$次多项式，使得 ?|$f(x)$|$1$ -|-|- $x=t$|$1$|$1$ $x=\lambda_1$|$1$|$1$ $x=\lambda_2$|$1$|$1$ $x=\lambda_3$|$1$|$1$ $\vdots$|$\vdots$| $x=\lambda_n$|$1$|$1$ $$ f(x)=1\cdot\frac{(x-\lambda_1)(x-\lambda_2)\cdots (x-\lambda_n)}{(t-\lambda_1)(t-\lambda_2)\cdots (t-\lambda_n)}+1\cdot\frac{(x-t)(x-\lambda_2)\cdots (x-\lambda_n)}{(\lambda_1-t)(\lambda_1-\lambda_2)\cdots (\lambda_1-\lambda_n)} $$ $$ +\cdots+1\cdot\frac{(x-t)(x-\lambda_1)\cdots (x-\lambda_{n-1})}{(\lambda_n-t)(\lambda_n-\lambda_1)\cdots (\lambda_n-\lambda_{n-1})} $$ 所以 $$ 1=1\cdot\frac{(x-\lambda_1)(x-\lambda_2)\cdots (x-\lambda_n)}{(t-\lambda_1)(t-\lambda_2)\cdots (t-\lambda_n)}+1\cdot\frac{(x-t)(x-\lambda_2)\cdots (x-\lambda_n)}{(\lambda_1-t)(\lambda_1-\lambda_2)\cdots (\lambda_1-\lambda_n)} $$ $$ +\cdots+1\cdot\frac{(x-t)(x-\lambda_1)\cdots (x-\lambda_{n-1})}{(\lambda_n-t)(\lambda_n-\lambda_1)\cdots (\lambda_n-\lambda_{n-1})} $$ #### 古典伴随矩阵 $\vec v_1,\vec v_2,\vec v_3\in \mathbb R^3$ $\det\begin{pmatrix}\vec v_1,&\vec v+\vec w,&\vec v_3\end{pmatrix}=\det\begin{pmatrix}\vec v_1,&\vec v,&\vec v_3\end{pmatrix}+\det\begin{pmatrix}\vec v_1,&\vec w,&\vec v_3\end{pmatrix}$ 现在假设我们要计算一个$3\times 3$矩阵的行列式 $$ \det\begin{pmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{pmatrix}= a_{11}\cdot\underbrace{\det\begin{pmatrix} 1&a_{12}&a_{13}\\ 0&a_{22}&a_{23}\\ 0&a_{32}&a_{33} \end{pmatrix}}_{A_{11}}+ a_{21}\cdot\underbrace{\det\begin{pmatrix} 0&a_{12}&a_{13}\\ 1&a_{22}&a_{23}\\ 0&a_{32}&a_{33} \end{pmatrix}}_{A_{21}}+ a_{31}\cdot\underbrace{\det\begin{pmatrix} 0&a_{12}&a_{13}\\ 0&a_{22}&a_{23}\\ 1&a_{32}&a_{33} \end{pmatrix}}_{A_{31}} $$