From 9c364a3f6c39e41be96e8961ad2557c2796e699c Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Fri, 26 Aug 2022 23:14:02 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- linear-algebra/exercise/3-vector/vector.pdf | Bin 78142 -> 82885 bytes linear-algebra/exercise/3-vector/vector.tex | 8 ++++++++ .../linear-equations-system.pdf | Bin 47304 -> 50279 bytes .../linear-equations-system.tex | 10 ++++++++++ linear-algebra/exercise/5-similar/similar.pdf | Bin 88858 -> 90856 bytes linear-algebra/exercise/5-similar/similar.tex | 14 +++++++++++--- .../linear-equations-system.pdf | Bin 178182 -> 121534 bytes .../linear-equations-system.tex | 2 ++ .../random-events-and-probability.pdf | Bin 127814 -> 127832 bytes .../random-events-and-probability.tex | 2 +- 10 files changed, 32 insertions(+), 4 deletions(-) diff --git a/linear-algebra/exercise/3-vector/vector.pdf b/linear-algebra/exercise/3-vector/vector.pdf index d560a1748627931fbd3e29a0d0bc6ca9cad3a13e..b962d7c3440d925a884fe9f4f3050e811c158f05 100644 GIT binary patch literal 82885 zcma&NV~nm#w}v}s+qP}nwr$(Cd$w)cwr$()ceZW&teu=6`#bx~$x13uo`02$x<}nt z)fh#pARC9W2j%SIWNK&&<*|8{-V?jU02A=S7otZ%IkO3-HYMOMJ|7)~Kpa1>h?fmEJ?triR zCpwUzR0Qj}CJHIaavf9r+>RB%CqPQE+A0rt(bvfXmNJoCKD*HYbMQ?_&zOA|5dTGp zuZ@rW%T2TK0#d+m_PZq&$7XcCfiReH0dx_Eu1%Y8R@0Sf3Vs=u*vFWV8yz%Xrx zw_B&idrBSAs5N#Y#mIX&q4bc2Xd*eLb(VXC2qUGu&v98!@}e|Uz2r;r21`_01dXgT zH=FLSu;X;**uN5Co*XEc)#ZI%F-5f2e=**-|Gr8)L`!}A!oe&&Pmi(6jfu3@PecdN z_7RrgdTvI%llQ<>^X8?4IjmB|yBjo3Md#DH%8Ds)>N3E+qXU{+28hjUH89Eea`F6v zZm5%E3niT!6cVTRyTXPv&;*{E2Y^js4>C!7VQCh$k5?M6K!5#k6~ zX`vqVudz&~-CcA6UVvEBpiJ#d{$J(&^Y=eeW&T&1WnyJy{14G$PuN|y-{sbw7V*N* z)T7y@#|EOxuFl5y8`6py8C}uZ3QYrgu2hoz#ZxuG;IM_{N1^JyN?T6 z1`()p+4)tEB%}Y$Q4spk<_NY8+5fqjh5i6s`zXpJu$KGTWp7CH^{1=!bVY9Cq?yee55ND-zI~WFyd>IQviaN{Mc` zTibM;6*8j~J`cF2KqtW!Uk9n>-FY;}!t>Y3Q? zP}&%hfC?6$+8=MU{5wC`>kTg@?|FvOlwt2(H{T^K{@%K0pXkANPO3f?36zspJ~IOB z*&f!(h%QUgQ|%J}@E z#9=#*L(-o3Xel^bF^X}u$P;7_X5BW`jlp*>(fWz~2@F$WBVIYxs{<0wz;K*-MbGfLD!zvVGydd*CmhMmgwQQsd?}8rXr7xB) z8b#HRS42pxQMgd$$}3zrDq@PrYY2gk%r2rDp~a*SujoLdSnD(nz0jbzdpKMNjy_~; zbJ>7JG)E?J%0TLJ05g$`#!Lb%SkKe2Mxxp8$S7>Z8+|B3wrh{lt{kk~N&x_p7<5To z2jJGQZ60rf(%tu%=QN5(00c-fU&eR@=NU6FJp!q9CY}K5_n^AtC=h8(pi`VfrtPUY z5z3k04uL#r0#IEp9_&hfXfK%*JB#C&IsBl)wyzwzr$EQ}veUes$8?YY# zXUQqL23@fK74MUA-!I*H_Pna2#(wfLE7jF^q*KU}&3PhoMaJIUgIZZw-=|U|$f`g# zzcseR)jQqowu_h80k4n}sl~_GfT_(NQ8{50$YTDTG9u&&${WDk5<~bD-&nxCS`&@f zkC-M5HOIxzjW5?1CfH<{RX@e=tW=wj(Uw3O)b8QQ)`arOi>N!*36q}pmgxHDUKL6R zWxNG~xbHhh_|;VTl+44H1eyEh=PHL?&W*%tS1Gm^v}zot3C>Y!WSr3nTHIvWGbqCO3loB zXeV46c{+4d5AoIPz=WA$gj?_Fl zq5xyJ?IR0KWbpj5H|SIEVpC=uf|G)UNdDBKj=x zW+ z7|PMqphJcv+^}DM9MIXfRYk&~bnq4y31_nfnI$BL(XsK;;Kmm*>5ehRD*n1GGs4c> z#5&esAMCg5;J}i|mVJBDgn=xvFKsdk&I8SkFm}1A5MDQc&&;q6FvLXcjJK!Je@~(} zm-vn_S#kqzD~Zmwq@~=fQK9+q4zr}Bns_CUgG>f3;qTkmgLs&%p4VV|#+GaQU@`KC z+ztt5d`9H_GT*S+-wvkLeUQ92Y}@iO-Dm_C*Sk;I#{Ot)(N{w(IY^-}Mt-xH5>QcEn^b3Lo!p%Q#bxJkvYidH$N_mGF~Sq!6~@TSjmWhE(x49r zDW-Uih+*m4)a7j2w<}v80LJW&7Qgnn?YjJ`o;KdK>|Jp1s$zkik(?#Sv}ByCd-92)fGr@rP-hOC7sT_e1nfBt(n#Slz1h*XB5wD2^v$xBs`s?L^O9KM*_NltAz+Z z=zvheamOYVgU=tY?YtG z7WAH~6X$mfQ3uAC6pk~M$j^0zi2w;5f~$@nXh1F)7|XyL3K!NTq*!6sIMthFIjlm` zJUgONK;~_>E{qRWs*mrwi00Ft(ec9io6{xp!il`T=hC=tz>w*qRGF z`$6UT-$;K&TMozJUS!MF-*<4}#V1?_iF{x`TB<~xgg+@_mhGq|-5QcEN{g9BKHx1V z;S-8C*Vz4Hoz3RG&ZSySFEvYuod`~bA19dLrO{K1-hRBcM-{DPh`cK5<&-Gbt0k$4 zm2_QP8#7m5vZQpg`76mu%^VN?fj5bH9%%&lZgW&2s+f0+a^ys~Uif(DR+tvo8zH9U zsqjsK$&ab}z1w%X8(@`3bo7&@Z{JROl-nmPC4v z^XEw2A~=03ZnJhl{HBmhDE%E?X{GNIe8K<^on5(chtffS&yax|dMc?Of!hz-mM7Rj z`**0n>^*#+evtWDdS)Ac8|!I+i_}lZqc)>RH}drT`~ZucD{VRVYd8}}ilKo1RP0Ts zUw}(UUcI5G1#WnDBhH+*dXEBd`Cfj{H@xy3ua?TUQGq-YT5>A zqFNV$Fdo@VKGNg3K~{RKxC}P&2o_d_FkiAyM99M{(fq?s z&G^@eQAWxeWf<8SS-4xi_9$exgW|v~h5C2Uz=((4>P{irFoVV^!>EP`1>eyf?|uosUX6 zl1U-GV>|(V6ZJJOl!v-9?vpOoR(WQLK7H)7MV9x$w>Qptajb6DDlVjcR5WweN3^Kb ztC6{#A$f66;gg)z<@|2nn>+j% zz#HyJ94Jl<>5|{3ITl-CfG5?5+t@)o*Hh-&)z0IeDaO{;;8gz7W3D{j96(YU8&$8S zD!?dClAe7+DpJY2uj2D1#iT>m)|f#R;kE&Au4jfH=}3=}DFSZqgH>^DE9ABBT*>v! z-FCN)0=jY+V&YRD3}tAHRpB9kPEfIUXo)lm;O?7y_N|p3_xg>$D6EFA=NVr82#aR; ztZZhIEEkl?PJ-wU_;Vm5pd-=#Fzwx8D+-i~Ez2(+LE8WhHO=l_5=y^=^DTgkb_(d8 zeuOk!d`t>J^LB+8M|`HVRav9!NX*N}DTO)Bg#k}nqx7c>x`mxHMqZ-IjAEN3+6a@w!qZ4@3#ip^Ntzfe29*4sg%;2`O
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zc#}(WO^q`D@s3f|p%HnIkK{z5B%Lg6RlmB=r@m+9o#9b1t5Sznwi{&1!0n1>MQT(JZOwO82YFOsNKpm52JDbUO=VP^lobaJ)xMGpr9~5 LJ3CbC75;w!2H|qR diff --git a/linear-algebra/exercise/5-similar/similar.tex b/linear-algebra/exercise/5-similar/similar.tex index 916cf92..6f4de26 100644 --- a/linear-algebra/exercise/5-similar/similar.tex +++ b/linear-algebra/exercise/5-similar/similar.tex @@ -157,16 +157,22 @@ $\left\{\begin{array}{l} \section{相似理论} +$P^{-1}AP=\Lambda$,$P$为特征向量组,$\Lambda$为特征值矩阵。 + \subsection{判断相似对角化} 可以使用相似对角化的四个条件,但是最基本的使用还是$A$有$n$个无关的特征向量$\xi$。 +判断以下条件即可相似对角化: + \begin{enumerate} - \item 判断是否为实对称矩阵,实对称矩阵必然相似于对角矩阵。 - \item 特征值是否都是实单根,相似于对角矩阵。 - \item 特征值为$n$重根,对应$n$个线性无关的特征向量,则相似于对角矩阵。如果小于则不相似。 + \item 实对称矩阵,即所有元素关于主对角线对称。 + \item 特征值都是实单根,即$n$个不同特征值,不存在重根。 + \item 特征值存在$t$重根,相同特征值对应$t$个线性无关的特征向量。(如果小于$n$则不相似) \end{enumerate} +一般都是第三种情况,判断存在重根后要使用$[\lambda E-A]$,然后计算$r(E-A)$,然后$s$自由变量值即无关特征向量值$=n-r$,如果$s=t$则可以相似对角化,如果$ssRShF$H2jM{Yb}^$fTk*9epvLnX$f+CF1?Cm;(~kGwp15yc9ukiALIiOR-rNkWN*iY7CS#}`F)zNY9^jM z?>;|2XKJQmdyIs4J1K1SMj*jEP|G~)cxCw?fe&DiDe9EffSnZbT0`vh-^qpjataRW z$o|>lvi8DiXISf~ixsQ@_@B5d@r`E?n-OGC6_8H6cfYsK#Vv*!^U%dkhahU@k{@$0 zX~LLuMb0&giMhjecFDSO7PE3Hyd9``5vxWXRHV|lfs`HeifK1YHa|# z^K&g|j_8|JxOVP)Ynpy__(D$lVtuM;&x?I?@zdwM*$YWAwel7x?+KpNX=lkt!Wd15 zsdF3P+t&-UiG?_aJ2eW-_b88+O6la(d&no|_ibn3zGQDaJ{_wln(NRlt1Uy9ntft^ z6lj;s3sCP`Pns`obr^Mnd-k&R556D+uV*nRE3bitn0{C$rU zj(Wx$Tf@1W8M@j`2Exf=!C%!mt0eMtC~u zJgt{Qa+P=9I{8H|4MA*9#5jmw>PiZKU1M z`VmKLQmy!njDo2Mw9-c*>nPyUL|wTLllmMKY}Gwe9u~<1wQ7zxn$r5Q)wZ0P=!wfe zxBw>B|Mli*OWTvMBX^&u|C<{YfR9u)iT7V#og~IFVCW;{`!RZ$d!pwCIZE~V`d+uR zWz}5POv;}i2|u8aQB52kg5NL(j>y&+=vkKbRp4K z#SLi{CUd`S8UpcU7$7?puy#mv+lEKB_Cp@0r$kpGX+FlZrDQjGxnnS8h1Q#>M-eBm zsWE>2`^u`Tj)rLNv%6@=>*t$Au)3$6k5-J_rWu`Qj9T~2H56JHUhh|+>IegsF)BA+)as*1V7d7a15@I= zkLa~X1Bv+ARu3qLweybM73btLvUo#jR%c}%A`lzLet9!Z!JRObtW9^B>wM}h!6df_ 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zFYd$rI2#Y(DqM|oa4xRFwU~_Sa0_n5dAJR?;~t!k1-KZO-~wETi|`;8V<8^G!*~Rb z<54_@XYd4`LNA`gB0P=f@hqOhn^=Y=cnPm!g^G7z=wFlKOlN7Rv+RGTvpm6WPL+x; z=t@phfGd8FmD?)yURN4ETd7fLbbqk~{#dD1X-aaXIn_#?N{io>whSwNmG(MU zI{x4PdX->>E1h4g+)?TJYx}NBPtbOw%E#~S0$,我们称在已知事件$A$发生的条件下,事件$B$发生的概率为\textbf{条件概率},记为$P(B|A)=\dfrac{P(AB)}{P(A)}$。$P(\overline{B}|A)=1-P(B|A)$,$P(B-C|A)=P(B|A)-P(BC|A)$。 \item 乘法公式:若$P(A)>0$,则$P(AB)=P(A)P(B|A)$。一般而言,对于$n>2$,$P(A_1A_2\cdots A_{n-1})>0$,则$P(A_1A_2\cdots A_{n-1})=P(A_1)P(A_2|A_1)P(A_3|A_1A_2)$\\$\cdots P(A_n|A_1A_2\cdots A_{n-1})$。($A_i$的顺序不定) \item 全概率公式:若$\bigcup\limits_{i=1}^nA_i=\Omega$,$A_iA_j=\varnothing$($i\neq j$,$i,j=1,2,\cdots,n$),$P(A_i)>0$,则对任一事件$B$,有$B=\bigcup\limits_{i=1}^nA_iB$,$P(B)=\sum\limits_{i=1}^nP(A_i)P(B|A_i)$。$P(B)=P(B\Omega)=P(B(\bigcup\limits_{i=1}^nA_i))=P(\bigcup\limits_{i=1}^nBA_i)=\sum\limits_{i=1}^nP(BA_i)=\sum\limits_{i=1}^nP(A_i)P(B|A_i)$。