4.矩陣的進(jìn)一步討論——相似與合同

一些簡(jiǎn)單的(半)正定問(wèn)題

例1.已知 $A=(a_{ij})$ 是 $n$ 級(jí)正定矩陣善玫，則
(1) $a_{ii}>0(i=1,2,\ldots,n))$
(2) $2|a_{ij}|<a_{ii}+a_{jj}(i\not=j)$
(3) $A$ 的所有元素中，絕對(duì)值最大元素一定在對(duì)角線上，從而也一定是正數(shù)
提示：取特殊的 $X$ 代入 $XAX'$

例2.已知 $a_1,a_2,\ldots,a_n$ 是兩兩互異的正數(shù)，則 $A=(\frac{1}{a_i+a_j})_{n\times n}$ 是正定矩陣
提示：計(jì)算行列式結(jié)果為 $|A|=\frac{\sum\limits_{i<j}(a_i-a_j)^2}{\sum\limits_{i,j}(a_i+a_j)}$

正定與半正定的 $C'C$

定理：(極其重要)設(shè) $A$ 是一個(gè) $m\times n$ 實(shí)矩陣怀樟，則r(A'A)=r(A)

例題1.證明：對(duì)實(shí)數(shù)域上的任意 $s\times n$ 實(shí)矩陣 $A$ 都有 $r(AA'A)=r(A)$
提示： $r(AA'A)\geq r(AA'AA')$

定理：(非常重要) $A$ 是任一 $s\times n$ 的實(shí)矩陣，則 $A'A$ 是半正定矩陣坡倔，并且 $A'A$ 是正定矩陣的充要條件是 $A$ 列滿(mǎn)秩漂佩。

命題：半正定手法：已知 $A$ 是 $n$ 級(jí)半正定矩陣脖含， $X$ 是一個(gè) $n$ 級(jí)實(shí)列向量罪塔，且 $X'AX=0$ 則 $AX=0$

例題2.已知 $A$ 是一個(gè) $n$ 級(jí)非對(duì)稱(chēng)的實(shí)矩陣，對(duì)任意的 $n$ 維實(shí)列向量 $X$ 都有 $X'AX\geq 0$ 养葵，且存在實(shí)向量 $\beta$ 使得 $\beta'A\beta=0$ 征堪，同時(shí)對(duì)任意的 $n$ 維實(shí)列向量 $a,b$ 當(dāng) $a'Ab\not=0$ 時(shí)，有 $a'Ab+b'Aa\not=0$ 关拒，證明對(duì)任意的 $n$ 維實(shí)列向量 $\eta$ 都有 $\eta'A\beta=0$

切換輸入中文和英文公式有點(diǎn)麻煩佃蚜，往后都采用英文進(jìn)行筆記書(shū)寫(xiě)，英文書(shū)寫(xiě)不規(guī)范的地方希望各位指出着绊，本人不勝感謝谐算。
It's a disdressing thing to exchange the input way of English and Chinese,I will take notes in English in the future.Please point out the not standardized translation,and I will be thankful for that.

example3.if $A=(a_{ij})_{n \times n}$ and $B=(b_{ij})_{n\times n}$ are real symmetric matrices of n-order,marked $C=(a_{ij}b_{ij})_{n\times n}$ ，then
(1)if both $A,B$ are semi positive definite,then $C$ is semi positive definite too.
(2)if $A,B$ are positive definite,then $C$ is positive definite too.

Tips: suppose $B=P'P$ and $P=(p_{ij})$ ,then $b_{ij}=\sum\limits_{k=1}^np_{ki}p_{kj}$
talking about $X'CX$

Applied of Orthogonal similarity and the orthogonal diagonalization of real symmetric matrices.

Theorem:All of the real symmetric matrices of n-order $A$ are orthogonal similarity to a diagonal matrix.

Example1.if $A$ is a real square of n-order,and the characteristic values are real,proof $A$ is a symmetric matrix if and only if $AA'=A'A$ .
Tips: $\Leftarrow$ :inductive method.

Example 2.
(1) $A$ is a real square of n-order,satisfied $AA'=A^2$ ,proof: $A$ is a symmetric square.
(2) $A$ is a real square of n-order,satisfied $A'A=A^2$ ,proof: $A$ is a symmetric square.
(3) $A$ is a real square of n-order,satisfied $AA'=-A^2(or A'A=-A^2)$ ,proof: $A$ is a antisymmetric square

Example 3.Proof:real matrix $A$ of n-order is orthogonal similar to a upper triangular matrix if and only if the characteristic value of $A$ are real

Example 4.Proof:All the complex matrix of n-order are similar to a upper triangular matrix.

Example 5.suppose $A,B$ are respectively $3\times 2$ and $2\times 3$ real matrix,and $AB,BA$ are symmetric,we know $AB=\left( \begin{array}{ccc} 8&2&-2\\ 2&5&4\\ -2&4&5 \end{array} \right)$ Proof:rank of $A,B$ are 2,calculate $BA$ .
Tips:
1.use the special value have a intuititive impress
2. $(AB)^2=9AB$
3.equivalent stantard type

Proof and application of quadratic inertia theorem.

(Quadratic inertia theorem)All of the quadratic which define on the real domain,can become a canonical shape after sevral non-degenerate linear replacements.And the canonical shape is unique.(The process of proof is classical)

Example.if the quadratic of n variables $f(X)=X'AX$ ,and $A$ is reversible,when $x_{k+1}=x_{k+1}=\ldots =x_n$ , $f(X)=0$ ,which $k\leq\frac{n}{2}$ ,proof the quadratic $f(X)$ symbol difference $t$ satisfied $|t|\leq n-2k$

Turn the matrix into a cannonical type or a standard type.

Example1.suppose real symmetric matrix $A$ ,sort all the characteristics values of it by size: $\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n$ proof:for any n dimensional vector $\alpha$ ,have $\lambda_n|\alpha|^2\leq\alpha'A\alpha \leq\lambda_1|\alpha|^2$

Example2.suppose $B$ is a real matrix of n-order,all the characteristcs values of $B'B$ are sorted by size $\lambda_1\geq \lambda_2\geq \ldots\geq\lambda_n\geq 0$ ,proof:any real characteristics value of $B$ marked $\mu$ satisified $\sqrt{\lambda_n}\leq|\mu|\leq\sqrt{\lambda_1}$

A special quadratic

Proposition:if $A=(a_{ij})_{s\times n}$ is a real matrix,then the n-variables quadratic $f(X)=\sum\limits_{i=1}^s(a_{i1}x_1+a{i2}x_2+\ldots+a_{in}xn)^2$ corresponding to the matrix $A'A$ .

Example.if $a_1,a_2,\ldots,a_n$ are real numbers,proof:n-variables quadratic $f(X)=(x_1+a_1x_2)^2+(x_2+a_2x_3)^2+\ldots+(x_{n-1}+a_{n-1}x_n)^2+(x_n+a_nx_1)^2$ is a positive definite quadratic if and only if $a_1a_2\ldots a_n\not=(-1)^n$

The condition of matrix diagonalizition

Example 1.we know $b_1,b_2,\ldots,b_n$ are positive real numbers,and $b_1+b_2+\ldots+b_n=1$ ,if $A=(a_{ij})$ ,which $\left\{\begin{array}{ccc} 1-b_j&&i=j\\ -\sqrt{b_ib_j}&&i\not=j\end{array} \right.$ please calculate the rank of $A$ ,and judge whether $A$ can be turn into a diagonalizition type,if can be diagonalizition,please write down the similar diagonalizition matrix of $A$ .

Some conclusions
1.if all of the characteristics value of a matrix is 0,then this matrix is power zero.
2.A matrix $A$ only have these characteristics values:0 and 1,we can't get the conclusion that $A$ is idempotent,but if this matrix is symmetric归露，we can get this conclusion.
3.if a real symmetry matrix only have these characteristics which values are $\pm1$ ,then $A$ is symmetry and orthogonal.

Example 2.Please calculate the determinant $\left|\begin{array}{cccccc} 1&2&3&\ldots&n-1&n\\ n&1&2&\ldots&n-2&n-1\\ n-1&n&1&\ldots&n-3&n-2\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 3&4&5&\ldots&1&2\\ 2&3&4&\ldots&n&1\end{array}\right|$
Tips: $J=\left(\begin{array}{cccc} 0&1&&\\ &\ldots&1&\\ &&\ldots&1\\ 1&&&0\end{array}\right)$ is similar to $diag(\omega,\omega^2,\ldots,\omega^{n-1},\omega^n)$

Idempoten matrix

Some character:
1. $A^2=A\Leftarrow\Rightarrow r(A)+r(A-E)=n$
2. $A$ is idempoten matrix,it can be turned into a diagonalization matrix.
3.if $A$ is a idempoten,then $r(A)=tr(A)$

Proposition
if $A_1,A_2,\ldots,A_m$ are square matrix of n-order,satisfied $A_1+A_2+\ldots+A_m=E_n$ then,these three thing are equivalance:
(1) $A_1,A_2,\ldots,A_m$ are all odempoten.
(2) $r(A_1)+r(A_2)+\ldots+r(A_m)=n$
(3) $\forall i\not=j(i,j=1,2,\ldots,n)$ we have $A_iA_j=0$

Antisymmetric matrix

Proposition
suppose $A$ is a antisymmetric matrix,then $A$ is contract to a quasi-diagonal matrix $diag\{\left(\begin{array}{cc} 0&1\\ -1&0\end{array}\right),\ldots,\left(\begin{array}{cc} 0&1\\ -1&0\end{array}\right),0,\ldots,0\}$

Smith orthogonalization

Theorem Any real reversible matrix can be decomposed into a orthogonal matrix and a upper triangular matrix whose element of digonal are positive number,and it has a unique decompose.

A general problem of similarity and contract

Example 1.if $A$ is a positive definite matrix of n-order, $B$ is a antisymmetry matrix,then $|A\pm B|>0$

Example 2.We know that $A,B$ are all real antisymmetry matrix,and $A$ is reversible,proof $|A^2-B|>0$

Tips: the antisymmetry matrix's characteristc values only can be 0 or a pure imaginary number.If a antisymmetry matrix $A$ is reversible then the $A^2$ must be a negative definite matrix,and the order of $A$ only can be even.

Important Proposition
We know that $A$ is a positive definite matrix of n-order,and $B$ is a semi-positive matrix of n-order,then $|A+B|\geq|A|+|B|$ ,and the eual sign only established when $B=0$

Proposition 2
$A,D$ are respectively $n-order$ , $m-order$ real matrix, $B$ is a $n\times m$ real matrix,and $\left(\begin{array}{cc} A&B\\ B'&D\end{array}\right)$ is a positive definite matrix.then
(1) $A,D,D-B'A^{-1}B,A-BD^{-1}B'$ are positive definite matrix.
(2) $\left|\begin{array}{cc} A&B\\ B'&D\end{array}\right|\leq|A||D|$ ,and the equals only established when $B=0$

Inference: $|AA'|\leq\prod\limits_{i=1}^n(a_{i1}^2+a_{i2}^2+\ldots+a_{in}^2)$ $(Hadamard\ inequality)$

Example 1.suppose $A,B$ are respectively $m\times n$ and $s\times n$ real row full rank matrix,marked $Q=AB'(BB')^{-1}BA'$ ,proof:
(1) $AA'-Q$ is semi-definite matrix.
(2) $0\leq|Q|\leq|AA'|$

Proposition 3
Suppose $A$ is a positive definite n-order matrix, $B$ is a real symmetry matrix of n-order,then exists reversible matrix $P$ such that $P'AP=E$ and $P'BP$ is a diagonal matrix.

Example 2.we know that $A$ is a positive definite matrix of n-order, $\alpha$ is a $n-dimension$ column vector,proof: $0\leq\alpha'(A+\alpha\alpha')^{-1}\alpha<1$
Tips: use the proposition 3.

Example 3.Suppose $A$ is a reversible real symmetry matrix of n-order,then $A$ is positive definite $\Leftarrow\Rightarrow$ $\forall$ n-order positive definite matrix $B$ ,have $tr(AB)>0$
Tips: if $A=T'\left(\begin{array}{ccc} \lambda_1&&\\ &\ldots&\\ &&\lambda_n\end{array}\right)T$ ,let $B=T'\left(\begin{array}{ccc} 1&&\\ &\ldots&\\ &&t\end{array}\right)T(t>0)$

Matrix equation $AX-XB=0$

Example 1.suppose $A,B$ are resectively n-order and m-order square on number of fields $\mathbb{P}$ ,proof:if both $A$ and $B$ has $r(0<r\leq min\{n,m\})$ common charateristc values which are not equal to any two of them,then the matrix equation $AX-XB=0$ has a matrix solution whose rank is $r$ .
Tips: if $A\xi_i=\lambda_i\xi_i$ and $B\eta_i=\lambda_i\eta_i$ ,let $C=\left(\begin{array}{cccc} \xi_1&\xi_2&\ldots&\xi_r\end{array}\right)\left( \begin{array}{c} \eta_1'\\ \eta_2'\\ \ldots\\ \eta_r'\end{array}\right)$

Example 2.suppose $A,B$ are respectively n-order and m-order matrix on the complex fields,proof:if the matrix equation $AX-XB=0$ has a matrix solution whose rank is $r$ ,then $A,B$ at least has $r$ common characteristics values(repeat roots are calculated as repeat count)

Tips we can write $X=P\left(\begin{array}{cc} E_r&0\\ 0&0\end{array}\right)Q$

Example 3.Suppose $A,B$ are n-order matrix,the characteristic polynomial of $A$ is $f(\lambda)$ ,proof: $f(B)$ is reversible $\Leftarrow\Rightarrow$ $A,B$ don't have common characteristic values

Tips: $\Rightarrow:$ contrary:suppose $A,B$ exists a common characteristc value $\lambda$ ,then $f(\lambda)=0$ ,and $f(\lambda)$ is the characteristic value of $f(B)$ ,then we know $f(B)$ is not reversible,we get a contradiction.

Matrix decomposition

Proposition1
$A$ is a n-order positive definite matrix,then
(1)exists a real reversible matrix $C_1$ such that $A=C_1'C_1$
(2)exists a real symmetry matrix $C_2$ such that $A=C_2^2$
(3)exists a positive definite matrix $C$ such that $A=C^2$ ,here $C$ is unique

Tips: $C$ is unique:we can write down $A=C^2=D^2$ ,and exists $T_1,T_2$ such that $C=T_1\left( \begin{array}{ccc} \sqrt{\lambda_1} E_1&&\\ &\ldots&\\ &&\sqrt{\lambda_s}E_s\end{array}\right)T_1^{-1}$ $D=T_2\left( \begin{array}{ccc} \sqrt{\lambda_1} E_1&&\\ &\ldots&\\ &&\sqrt{\lambda_s}E_s\end{array}\right)T_2^{-1}$

Proposition 2
$A,B$ are two n-order matrix on real fields,and $B$ is a semi-definite matrix, $m$ is a positive integer,satisfied $AB^m=B^mA$ ,then $AB=BA$

Example 1. $A$ is a real symmetry matrix of n-order.proof:exists a real symmetry matrix $B$ such that $A=B^2$ if and only if $A$ is a semi-definite matrix.

Example 2.for any real reversible matrix $A$ ,there must exists a orthogonal matrix $T$ and two positive definite matrix $S_1,S_2$ ,such that $A=TS_1=S_2T$ ,and these two decomposition are unique.
Tips: $A'A=S_1^2\Rightarrow A=(A')^{-1}S_1^2$ let $T=(A')^{-1}S_1$ ,obviously $TT'=(A')^{-1}S_1S_1'A^{-1}=A'^{-1}A'AA^{-1}=E$ then, $T$ is orthogonal,and using the knowledge of last example,we know $S_1$ is positive definite,in this way, $A=(A')^{-1}S_1A'(A')^{-1}S_1=(A')^{-1}S_1A'T$ let $S_2=(A')^{-1}S_1A'$ then $A=TS_1=S_2T$
unique:suppose exists different $\hat{T}$ and $\hat{S_1}$ such that $A=TS_1=\hat{T}\hat{S_1}$ then $A'A=S_1'T'TS_1=S_1'S_1=\hat{S_1'}\hat{S_1}$ we use the proposition 1 know that $S_1$ is unique

Example 3.proof:for any reversible matrix $A$ of n-order,there exists orthogonal matrix $T_1,T_2$ such that $A=T_1diag\{\lambda_1,\lambda_2,\ldots,\lambda_n\}T_2$ ,which $\lambda_1^2,\lambda_2^2,\ldots,\lambda_n^2$ are all the characteristic values of $A'A$ ,and $\lambda_i>0(i=1,2,\ldots,n)$

最后編輯于：2018.09.26 12:26:49

?著作權(quán)歸作者所有,轉(zhuǎn)載或內(nèi)容合作請(qǐng)聯(lián)系作者

人面猴
序言：七十年代末洲脂，一起剝皮案震驚了整個(gè)濱河市，隨后出現(xiàn)的幾起案子剧包，更是在濱河造成了極大的恐慌恐锦，老刑警劉巖，帶你破解...
沈念sama閱讀 222,104評(píng)論 6贊 515
死咒
序言：濱河連續(xù)發(fā)生了三起死亡事件疆液，死亡現(xiàn)場(chǎng)離奇詭異一铅，居然都是意外死亡，警方通過(guò)查閱死者的電腦和手機(jī)堕油，發(fā)現(xiàn)死者居然都...
沈念sama閱讀 94,816評(píng)論 3贊 399
救了他兩次的神仙讓他今天三更去死
文/潘曉璐我一進(jìn)店門(mén)潘飘，熙熙樓的掌柜王于貴愁眉苦臉地迎上來(lái)，“玉大人掉缺，你說(shuō)我怎么就攤上這事福也。” “怎么了攀圈？”我有些...
開(kāi)封第一講書(shū)人閱讀 168,697評(píng)論 0贊 360
道士緝兇錄：失蹤的賣(mài)姜人
文/不壞的土叔我叫張陵暴凑，是天一觀的道長(zhǎng)。經(jīng)常有香客問(wèn)我赘来，道長(zhǎng)现喳，這世上最難降的妖魔是什么凯傲？我笑而不...
開(kāi)封第一講書(shū)人閱讀 59,836評(píng)論 1贊 298
?港島之戀（遺憾婚禮）
正文為了忘掉前任，我火速辦了婚禮嗦篱，結(jié)果婚禮上冰单，老公的妹妹穿的比我還像新娘。我一直安慰自己灸促，他們只是感情好诫欠，可當(dāng)我...
茶點(diǎn)故事閱讀 68,851評(píng)論 6贊 397
惡毒庶女頂嫁案：這布局不是一般人想出來(lái)的
文/花漫我一把揭開(kāi)白布。她就那樣靜靜地躺著浴栽，像睡著了一般荒叼。火紅的嫁衣襯著肌膚如雪。梳的紋絲不亂的頭發(fā)上典鸡，一...
開(kāi)封第一講書(shū)人閱讀 52,441評(píng)論 1贊 310
城市分裂傳說(shuō)
那天被廓，我揣著相機(jī)與錄音，去河邊找鬼萝玷。笑死嫁乘，一個(gè)胖子當(dāng)著我的面吹牛，可吹牛的內(nèi)容都是我干的球碉。我是一名探鬼主播蜓斧，決...
沈念sama閱讀 40,992評(píng)論 3贊 421
雙鴛鴦連環(huán)套：你想象不到人心有多黑
文/蒼蘭香墨我猛地睜開(kāi)眼，長(zhǎng)吁一口氣：“原來(lái)是場(chǎng)噩夢(mèng)啊……” “哼睁冬！你這毒婦竟也來(lái)了挎春？” 一聲冷哼從身側(cè)響起，我...
開(kāi)封第一講書(shū)人閱讀 39,899評(píng)論 0贊 276
萬(wàn)榮殺人案實(shí)錄
序言：老撾萬(wàn)榮一對(duì)情侶失蹤痴突，失蹤者是張志新（化名）和其女友劉穎搂蜓，沒(méi)想到半個(gè)月后，有當(dāng)?shù)厝嗽跇?shù)林里發(fā)現(xiàn)了一具尸體辽装，經(jīng)...
沈念sama閱讀 46,457評(píng)論 1贊 318
?護(hù)林員之死
正文獨(dú)居荒郊野嶺守林人離奇死亡帮碰，尸身上長(zhǎng)有42處帶血的膿包…… 初始之章·張勛以下內(nèi)容為張勛視角年9月15日...
茶點(diǎn)故事閱讀 38,529評(píng)論 3贊 341
?白月光啟示錄
正文我和宋清朗相戀三年，在試婚紗的時(shí)候發(fā)現(xiàn)自己被綠了拾积。大學(xué)時(shí)的朋友給我發(fā)了我未婚夫和他白月光在一起吃飯的照片殉挽。...
茶點(diǎn)故事閱讀 40,664評(píng)論 1贊 352
活死人
序言：一個(gè)原本活蹦亂跳的男人離奇死亡，死狀恐怖拓巧，靈堂內(nèi)的尸體忽然破棺而出斯碌，到底是詐尸還是另有隱情，我是刑警寧澤肛度，帶...
沈念sama閱讀 36,346評(píng)論 5贊 350
?日本核電站爆炸內(nèi)幕
正文年R本政府宣布傻唾，位于F島的核電站，受9級(jí)特大地震影響，放射性物質(zhì)發(fā)生泄漏冠骄。R本人自食惡果不足惜伪煤，卻給世界環(huán)境...
茶點(diǎn)故事閱讀 42,025評(píng)論 3贊 334
男人毒藥：我在死后第九天來(lái)索命
文/蒙蒙一、第九天我趴在偏房一處隱蔽的房頂上張望凛辣。院中可真熱鬧抱既，春花似錦、人聲如沸扁誓。這莊子的主人今日做“春日...
開(kāi)封第一講書(shū)人閱讀 32,511評(píng)論 0贊 24
一樁弒父案，背后竟有這般陰謀
文/蒼蘭香墨我抬頭看了看天上的太陽(yáng)蝗敢。三九已至捷泞，卻和暖如春，著一層夾襖步出監(jiān)牢的瞬間前普，已是汗流浹背肚邢。一陣腳步聲響...
開(kāi)封第一講書(shū)人閱讀 33,611評(píng)論 1贊 272
情欲美人皮
我被黑心中介騙來(lái)泰國(guó)打工壹堰，沒(méi)想到剛下飛機(jī)就差點(diǎn)兒被人妖公主榨干…… 1. 我叫王不留拭卿，地道東北人。一個(gè)月前我還...
沈念sama閱讀 49,081評(píng)論 3贊 377
代替公主和親
正文我出身青樓贱纠，卻偏偏與公主長(zhǎng)得像峻厚，于是被迫代替她去往敵國(guó)和親。傳聞我的和親對(duì)象是個(gè)殘疾皇子谆焊，可洞房花燭夜當(dāng)晚...
茶點(diǎn)故事閱讀 45,675評(píng)論 2贊 359