高数线代常用结论长更页
I would like to give some useful features widely used in Advanced Algebra und Analysis but rarely in Engineer. From the outset, I will list down those shit in the first part of this blog, and then give the proof of some of them depending on my mood. I hope this blog would being helpful.
代数学部分
正定矩阵的一些性质
设 (A) 为 (p) 阶对称矩阵，(x)是一个(p)维向量，则(x^TAx)称为(A)的二次型，for (forall x eq0), (x^TAx>0(geq0))，则称(A)为正定(非负定)矩阵，记作 (A>0(geq0))；
- 设(A)为对称矩阵，则(A)是正定矩阵，当且仅当(A)的所有特征值均为正；
- if (A>0), then (A^{-1}>0);
- if (A>0), there exist a matrix (A^{frac12}>0), s.t. (A=A^{frac12}A^{frac12}).
- If (A={(a_{ij})_{p imes p}|rank(A)=r,\,Ageq0}), there is a matrix (B_{p imes r}) for whose rank is (r), derives (A=BB^T).
Wallis 公式及积分公式的推广
[int_0^{2pi}(sin{x})^ndx=
left{
egin{array}{c}
4int_0^{fracpi2}(sin{x})^ndx&n为偶数时\
0&n为奇数时
end{array}
ight.
]
上式中 (sin x) 可以替换为 (cos x)。
[int_0^{pi/2}(sin{x})^ndx=
left{
egin{array}{c}
frac{n-1}{n}frac{n-3}{n-2}cdotsfrac{2}{3}cdot1&n为奇数时\
frac{n-1}{n}frac{n-3}{n-2}cdotsfrac{1}{2}cdotfracpi2&n为偶数时
end{array}
ight.
]
上式即为著名的 Wallis公式
[int_0^{pi}(sin{x})^ndx=2int_0^{pi/2}(sin{x})^ndx=2int_0^{pi/2}(cos{x})^ndx
]
[int_0^{pi}xf(sin{x})dx=fracpi2int_0^{pi/2}f(sin{x})dx=fracpi2int_0^{pi/2}f(cos{x})dx
]
通过上述两个式子，可以将复杂的含三角函数的定积分转化为可以通过沃里斯公式计算的形式。
通过定积分定义计算 n 项和、积的极限
[lim_{n oinfty}frac1nsum_{i=1}^{kn}f(frac{i}{n})=int_{0}^kf(x)dx
]
[lim_{n oinfty}frac{b-a}nsum_{i=1}^{n}fleft[a+(b-a)frac{i}{n}
ight]=int_a^bf(x)dx
]
证明
日后再更，爷懒了