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家里蹲大学数学杂志按学校分类目录[2016年8月23日更新]

├─中南大学│      第3卷第81期_中南大学2011年数学分析考研试题参考解答│      第4卷第266期_中南大学2013年高等代数考研试题参考解答│      ├─中国人民大学│      第5卷第370期_中国人民大学2003年高等代数考研试题参考解答│      第5卷第371期_中...

Dos操作

\tree/f >c.txt   \dir/s/b >c.txt

中文的习题解答中国人看懂, 英文的习题解答外国人能看懂

中文的习题解答中国人看懂, 英文的习题解答外国人能看懂. 接到好几个老外的邮件了...

[再寄小读者之数学篇](2014-11-26 广义 Schur 分解定理)

设 $A,B\in \bbR^{n\times n}$ 的特征值都是实数, 则存在正交阵 $P,Q$ 使得 $PAQ$, $PBQ$ 为上三角阵.

[再寄小读者之数学篇](2014-11-26 正定矩阵乘积的特征值)

设 $A,B$ 都是实正定矩阵, 则 $A^{-1}B$ 的特征值都是正实数.

[再寄小读者之数学篇](2014-11-26 分块矩阵求逆)

如果 $A$ 可逆或 $D$ 可逆, 则 $$\bex \sev{\ba{cc} A&B\\ C&D \ea}=|A|\cdot |D-CA^{-1}B| =|D|\cdot |A-BD^{-1}C|. \eex$$

[再寄小读者之数学篇](2014-11-26 幂等矩阵的一个充分条件)

若 $A\in \bbR^{m\times n}$ 列满秩, 则 $A(A^TA)^{-1}A^T$ 是幂等矩阵, 其特征值为 $1$ 或 $0$, 且存在正交阵 $Q$, 使得 $$\bex Q^T[A(A^TA)^{-1}A^T]Q=\sex{E_n\atop 0}. \eex$$

2014年11月21日下午游

人生无常

今闻高二同桌悄然去世. 而前几年已有一位同学去世.   高中也不知道好不好, 有个高考, 让人有了机会出去. 但为此, 我却付出了 "腰椎间盘突出" 的苦难. 高中毕业在家自己还以为是肾有问题...

[再寄小读者之数学篇](2014-11-24 积分中值定理)

积分第一中值定理. 若 $f$ 在 $[a,b]$ 上连续, 则 $$\bex \exists\ \xi\in (a,b),\st \int_a^b f(x)\rd x=f(\xi)(b-a). \eex$$ 推广的积分第一中值定理.

[再寄小读者之数学篇](2014-11-24 Abel 定理)

设幂级数 $\dps{g(x)=\sum_{n=0}^\infty a_nx^n}$ 在 $|x|N\ra |s_k-s|

SCI杂志分区规则

1区:该期刊的影响因子排名位于其所在学科排名的前5% 2区:该期刊的影响因子排名位于其所在学科排名的前20%但未进入5% 3区:该期刊的影响因子排名位于其所在学科排名的前50%但未进入20%的 4区:该期刊的影响因子排名位于其所在学科排名的50%

玉米真好吃

四川又发生地震了

四川康定发生地震了. 雅安, 汶川. 多灾多难的人...活着真的不容易.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]PrI.6.1

Given a basis $U=(u_1,\cdots,u_n)$ not necessarily orthonormal, in $\scrH$, how would you compute the biorthogonal basis $\sex{v_1,\cdots,v_n}$? Find ...

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.10

Every $k\times k$ positive matrix $A=(a_{ij})$ can be realised as a Gram matrix, i.e., vectors $x_j$, $1\leq j\leq k$, can be found so that $a_{ij}=\sef{x_i,x_j}$ for all $i,j$.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.9

(Schur's Theorem) If $A$ is positive, then $$\bex \per(A)\geq \det A. \eex$$   Solution. By Exercise I.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.8

Prove that for any matrices $A,B$ we have $$\bex |\per (AB)|^2\leq \per (AA^*)\cdot \per (B^*B). \eex$$ (The corresponding relation for determinants is an easy equality.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.7

Prove that for any vectors $$\bex u_1,\cdots,u_k,\quad v_1,\cdots,v_k, \eex$$ we have $$\bex |\det(\sef{u_i,v_j})|^2 \leq \det\sex{\sef{u_i,u_j}}\cdot...

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.6

Let $A$ be a nilpotent operator. Show how to obtain, from aJordan basis for $A$, aJordan basis of $\wedge^2A$.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.5

Show that the inner product $$\bex \sef{x_1\vee \cdots \vee x_k,y_1\vee \cdots\vee y_k} \eex$$ is equal to the permanent of the $k\times k$ matrix $\sex{\sef{x_i,y_j}}$.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.4

If $\dim \scrH=3$, then $\dim \otimes^3\scrH =27$, $\dim \wedge^3\scrH =1$ and $\dim \vee^3\scrH =10$.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.3

Let $\scrM$ be a $p$-dimensional subspace of $\scrH$ and $\scrN$ its orthogonal complement. Choosing $j$ vectors from $\scrM$ and $k-j$ vectors from $...

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.2

The elementary tensors $x\otimes \cdots \otimes x$, with all factors equal, are all in the subspace $\vee^k\scrH$.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.1

Show that the inner product $$\bex \sef{x_1\wedge \cdots \wedge x_k,y_1\wedge \cdots\wedge y_k} \eex$$ is equal to the determinant of the $k\times k$ matrix $\sex{\sef{x_i,y_j}}$.

[再寄小读者之数学篇](2014-11-21 关于积和式的一个不等式)

在 Rajendra Bhatia 的 Matrix Analysis 中, Exercise I.5.8 说: Prove that for any matrices $A,B$ we have $$\bex |\per (AB)|^2\leq \per (AA^*)\cdot \per (B^*B).

积和式Permanent在Mathematica中的运算

Permanent[m_List] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v]]   参考资料: http://mathworld.wolfram.com/Permanent.html

[再寄小读者之数学篇](2014-11-20 计算二重积分)

(from X.L. Zhen) 计算二重积分 $$\bex \iint_{\bbR^2}e^{-(x^2+xy+y^2)}\rd x\rd y. \eex$$   解答: $$\beex \bea \iint_{\bbR^2}e^{-(x^2+xy+y^2)}\rd x\rd y &=\iin...

expunge

expunge 擦掉； 除去； 删去； 消除   1. The experience was something he had tried to expunge from his memory. 他曾努力将那段经历从记忆中抹去。

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.4.6

Let $A$ and $B$ be two matrices (not necessarily of the same size). Relative to the lexicographically ordered basis on the space of tensors, the matri...

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.4.5

Suppose it is known that $\scrM$ is an invariant subspace for $A$. What invariant subspaces for $A\otimes A$ can be obtained from this information alone?   Solution.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.4.4

(1). There is a natural isomorphism between the spaces $\scrH\otimes \scrH^*$ and $\scrL(\scrH,\scrK)$ in which the elementary tensor $k\otimes h^*$co...

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.4.1

Let $x,y,z$ be linearly independent vectors in $\scrH$. Find a necessary and sufficient condition that a vector $w$ mush satisfy in order that the bil...

[再寄小读者之数学篇](2014-11-19 $\sin(x+y)=\sin x\cos y+\cos x\sin y$)

$$\bex \sin(x+y)=\sin x\cos y+\cos x\sin y. \eex$$ Ref. [Proof Without Words: Sine Sum Identity, The College Mathematics Journal].

[再寄小读者之数学篇](2014-11-19 等差数列的部分和)

设 $\sed{a_k}_{k=1}^n$ 为等差数列, 则 $$\bex a_1+\cdots+a_n=\frac{n(a_1+a_n)}{2}. \eex$$ Ref. [Proof Without Words: Partial Sums of an Arithmetic Sequence, The College Mathematics Journal].

[再寄小读者之数学篇](2014-11-19 $\sin x/x$ 在 $(0,\pi/2)$ 上递增)

$$\bex \frac{\sin x}{x}\nearrow. \eex$$ Ref. [Proof Without Words: Monotonicity of $\sin x/x$ on $(0,\pi/2)$, The College Mathematics Journal]

[再寄小读者之数学篇](2014-11-19 $\tan x/x$ 在 $(0,\pi/2)$ 上递增)

$$\bex \frac{\tan x}{x}\nearrow. \eex$$ Ref. [Proof Without Words: Monotonicity of $\tan x/x$ on $(0,\pi/2)$, The College Mathematics Journal].

[再寄小读者之数学篇](2014-11-19 一个代数不等式)

$$\bex \sqrt{x^2+x+1}+ \sqrt{y^2+y+1} +\sqrt{x^2-x+1}+ \sqrt{y^2-y+1}\geq 2(x+y). \eex$$ Ref. [Proof Without Words: An Algebraic Inequality, The College Mathematics Journal].

[再寄小读者之数学篇](2014-11-19 关于平方数的交叉和的两个代数等式)

For $n\geq 1$ to be an integer, $$\bex (2n)^2-(2n+1)^2+\cdots+(4n)^2 =-(4n+1)^2+\cdots+(6n)^2, \eex$$ $$\bex (2n+1)^2-(2n+2)^2+\cdots+(4n-1)^2 =-(4n)^2+(4n+1)^2-\cdots+(6n-1)^2.

[再寄小读者之数学篇](2014-11-14 矩阵的应用: 有限几何)

每个有限几何的线的条数 $\geq$ 点的个数. 若一个有限几何的线数 $=$ 点数, 则任意两条线都相交.

[再寄小读者之数学篇](2014-11-14 矩阵的应用: 多项式)

多项式 $$\bex p(z)=z^n+a_{n-1}x^{n-1}+\cdots+a_0 \eex$$ 的根的估计.

[再寄小读者之数学篇](2014-11-14 矩阵的应用: 代数)

Hilbert 零点定理: 设 $\bbF$ 是一个代数闭域, $L$ 是 $\bbF[x_1,\cdots,x_n]$ 的一个真理想, 则 $$\bex \exists\ (a_1,\cdots,a_n)\in\bbF^n\ra f(a_1,\cdots,a_n)=0,\quad\forall\ f\in L.

[再寄小读者之数学篇](2014-11-14 矩阵的应用: 数论)

1. 代数数: $\al\in\bbC$ 称为代数数, 如果它是某个系数为有理数的非零多项式的根. 2. 代数数全体构成一个域. (利用伙伴矩阵, 张量积很容易证明) 3. 代数整数: $\al\in\bbC$ 称为代数整数, 如果它是某个首一整系数多项式的根.

[再寄小读者之数学篇](2014-11-14 矩阵的应用: 友谊定理)

友谊定理: 如果在一群人中任何两个人都恰好有一个共同的朋友, 那么有一个人是每个人的朋友.

[再寄小读者之数学篇](2014-11-14 正整数的平方根要么为无理数, 要么为整数)

$$\bex \forall\ m\in \bbZ^+\ra \sqrt{m}\in (\bbR\bs \bbQ)\cup \bbZ^+. \eex$$

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.3.7

For every matrix $A$, the matrix $$\bex \sex{\ba{cc} I&A\\ 0&I \ea} \eex$$ is invertible and its inverse is $$\bex \sex{\ba{cc} I&-A\\ 0&I \ea}.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.3.6

If $A$ is a contraction, show that $$\bex A^*(I-AA^*)^{1/2}=(I-A^*A)^{1/2}A^*. \eex$$ Use this to show that if $A$ is a contraction on $\scrH$, then t...

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.3.1

Let $A=A_1\oplus A_2$. Show that   (1). $W(A)$ is the convex hull of $W(A_1)$ and $W(A_2)$; i.e., the smallest convex set containing $W(A_1)\cup W(A_2)$.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.10

(1). The numerical radius defines a norm on $\scrL(\scrH)$.   (2). $w(UAU^*)=w(A)$ for all $U\in \U(n)$.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.2.9

(1). When $A$ is normal, the set $W(A)$ is the convex hull of the eigenvalues of $A$. For nonnormal matrices, $W(A)$ may be bigger than the convex hull of its eigenvalues.