Linear Algebra II (Lecture)

Hold by Prof. Matheusz Michałek at theUniKonstanz during the2nd_Semester.

Recap

11

11.1 Scalar Produdct

• def. for $\mathbb{C}$ prove them all
• $|⟨v,w⟩{|}^2\le ⟨v,v⟩⟨w,w⟩$ "$\sim$" $(xy{)}^2=xx\cdot yy$ .8
• $\cos (\alpha )=\frac{⟨v,w⟩}{||v||\cdot ||w||}$

11.2

• $U^{\perp }:=\{v\in V|{\forall }_{u\in U}u\perp v\}$
• for $U$ subsp. of $V$, there’s maximally one $w\in U$ s.t. $v-w\in U^{\perp }$, $w$ is the orth. proj. of $v$ on $U$
• Gram-Schm. want $(w_i{)}_{i\in I}$ $W_1=v_1$, $w_2=v_2-\frac{⟨v_2,w_1⟩}{|w_1^2|}{w}_1$, $w_3=v_3-\frac{⟨v_3,w_1⟩}{|w_1{|}^2}{w}_1-\frac{⟨v_3,w_2⟩}{|w_2{|}^2}{w}_2$.
• $dim(U)+dim(U^{\perp })=dim(V)$
• $A$ is orth. $\iff row(A)$ is ONB $\iff column(A)$ is ONB $\iff A^{\ast }A=I_n$ (${}^{\ast }$ conj. trans.) .27

11.3 Hermitian Matrix

• $⟨f(v)w⟩=⟨v,f(w)⟩\iff f$ is self adjoint
• $A$ selfadj. $\iff A^{\ast }=A$, in $S\mathbb{R}$ or hermitian in $\mathbb{C}$.
• $A$ selfadj. $\Rightarrow \lambda \in \mathbb{R}$ for any eigenval.
• $f$ selfadj. and $V\to V$ the eigenvec. of $f$ make an ONB, also $f$ diagonalisable .9.
• $A$ selfadj. $A=P^{\ast }DP$ for $D$ diag., $P$ orth. ($\Rightarrow A\approx D$)

12

12.1 inf. sup, min. max

• Partial Ordering: reflexive, antisymmetrical and transitive
• $a\in ub(B)\iff {\forall }_{b\in B}a⪯b$, $sup(B)$ is the smallest such $a$; $a=max(B)\iff a\in B\cap ub(B)$.

12.2 Zorn’s Lemma

• $a$ maximal element in $B$ $\iff b\in B\wedge \neg {\exists }_{c\in B}b\prec c$.
• A chain of $A$ is a totally ordered subset of $A$
• Zorn’s Lemma: let $(A,⪯)$ s.t. any chain in $A$ has an $ub$ then $A$ has a maximal element.
• Cor. every vectorspace has a basis & every ideal is contained in a maximal ideal.

13

13.1 Dualspace

• $V^{\ast }:=Hom(V,\mathbb{K})$, dualbasis $v_i^{\ast }(v_i)=1$ and all the rest to $0$.
• $({\mathbb{K}}^n{)}^{\ast }\cong {\mathbb{K}}^n$, .6 for the function
• $f^{\ast }:l\mapsto l\circ f$ and note: $(f\circ g{)}^{\ast }=g^{\ast }\circ f^{\ast }$, also $h:f\mapsto f^{\ast }$, $h$ is lin.
• $M(f^{\ast },w^{\ast },v^{\ast })=M(f,w,v{)}^T$, “Transponieren heißt Komponenten dualisieren.”
• $f:V\to W$: $dim(ker(f))+dim(im(f^{\ast }))=dim(V)$, $dim(im(f))+dim(ker(f^{\ast }))=dim(W)$
• $f$ inj. $\iff f^{\ast }$ surj. and vice versa
• dualdual?

13.2 Bilinear Function

• $b(\cdot ,v_2)$ and $b(v_1,\cdot )$ linear, $b(v_1,v_2)\in W$ for $v_1\in V_1$ and $v_2\in V_2$. Too general

13.3 Bilinear Forms

• $V\times V\to K$, denoted with $Bil(V)$, Darst. $(M(b,\underline{v}){)}_{i,j}=b(v_i,v_j)$
• $b^{\to }:v\mapsto b(\cdot ,v)$, it fixes first the right variable, $b^{\leftarrow }$ does the opposite.
• $M(b,\underline{v})=M(b^{\leftarrow },\underline{v},{\underline{v}}^{\ast })=M(b^{\to },\underline{v},{\underline{v}}^{\ast }{)}^T$, and $b^{\to }=(b^{\leftarrow }{)}^{\ast }$
• $b$ not deg. $\iff b^{\to }$ inj. $\Rightarrow ker(b^{\to })=0\Rightarrow (b(\cdot ,v)=0)\to v=0$.

13.4 Quadratical Forms

• Quad forms $\cong$ Symmatrix $\cong$ Symm bilin. forms,
• see 13.4.6, $b(v,v)$ converse is $\frac{1}{2}(q(v+w)-q(v)-q(w))$

13.5 Cholesky

• See the scritp, remember the form of $D$ and $P$ and the ways to reform the quadratic form.
• in $P$ there can be blocks $\{\{1,1\},\{1,-1\}\}$ with correspondance of $\{\{\lambda ,0\},\{-\lambda ,0\}\}$ in $D$
• Cor. (not from above since $P$ has no need to be orth.) $A$ sym. $\Rightarrow A$ diag.

14

14.1 Sylvester

• Sylv is only on $Q(V)$, the quadratical forms
• .2.a number of pos. and neg. Eigenval.
• .2.c compute Cholesky and check the sign of elements in $D$ which are the $\lambda$.
• Mail Mario: if no Hauptmin. $=0$, then Vorzeichnenwechsel of them is $r$, $(n-r,r)$.

14.2 spd

• for $q\in Q(V)$ in an $\mathbb{R}$-VS, we say $spd(q)\iff {\forall }_{\bar{a}\in V}q(\bar{a})\ge 0$, $>$ for $pd(q)$.
• Cholesky only on $spd(A)$, with $p_{ii}=1$ and $d_{ii}>0$ .8
• $spd(q)\iff Sylv(q)=(r,0)\iff {\exists }_{l_1,...,l_n\in V^{\ast }}q(v)={\sum }_{i=1}^n{l}_i^2(v)$ .7
• $pd(q)\iff Sylv(q)=(n,0)$ (n is the max) $\iff$ there’s a basis of $V^{\ast }$, $l_1,...,l_n$, $q(v)={\sum }_{i=1}^n{l}_i^2(v)$
• $spd(A)\iff$ all real coef. of ${\chi }_A(-X)$ are $\ge 0$ (ez, not zerfallen) $\iff A=B^{T}B$ (in any way) $\iff ⟨\cdot ,\cdot ⟩$
• $pd(A)\iff$ all Leithau. $>0\iff$ has a Cholesky with $D>0$.
• $pd(q_1)$ then there’s $\underline{v}$ s.t. $M(q_1,\underline{v}),M(q_2,\underline{v})$ are diagonal.

15

15.1 Adjoint

• $⟨f(v),w⟩=⟨v,f^{\ast }(w)⟩$, if $dim(V)\ge \omega$ see 15.1.1, else $f^{\ast }:W\to V$ for $f:V\to W$, linear 15.1.6
• if $f$ in ${\mathbb{R}}^n$ then $f^{\ast }=f^{ad}$, and $M(f^{\ast },\underline{w},\underline{v})=M(f,\underline{w},\underline{v}{)}^{\ast }$, conj. trans. (hermitian don’t change)
• For also inf. VS: $ker(f^{\ast })=(im(f){)}^{\perp }$, $f$ ismorphism $\iff f^{\ast }=f^{-1}$ 15.1.12

15.2 Normal Functions

• $f$ normal $\iff f\circ f^{\ast }=f^{\ast }\circ f$, of $f$ and $f^{\ast }$ eigenvec. are the same and eigenval. are conj.
• Take an ONB of $V$ and let $f$ be the function with those eigenvec. then $f$ is normal .4
• There is a $\underline{v}$ ONB s.t. $M(f,\underline{v})$ is diag. $\iff f$ normal
• $A$ orth. $\iff$ exists $\underline{v}$ ONB s.t. $M(f,\underline{v})$ diagonal with every $|{\lambda }_i|=1$.
• $f$ orth. is a VS with $⟨\cdot ,\cdot ⟩$ isomorphism in $\mathbb{R}$ and $f$ unitary in $\mathbb{C}$, $⟨f(v),f(w)⟩=⟨v,w⟩$ 11.2.19
• $R\in {\mathbb{R}}^{n\times n}$ is orth. $\iff$ $R^T=R^{-1}$; and $C\in {\mathbb{C}}^{n\times n}$ is unitary $\iff C^{\ast }=C^{-1}$ also: (ext, and ext)
• ext: for $C\in {\mathbb{C}}^{n\times n}$: $C$ normal $\Rightarrow C$ diag.; $(C^{\ast }=C^{-1})\vee (C=C^{\ast })\vee (C^{\ast }=-C)\Rightarrow C$ normal
• blue table:
  • $f$ orth. ($\iff {\forall }_{v\in V}||f(v)||=||v||$) ONB with eigenval on the unit circle, $|{\lambda }_i|=1$
  • $f$ selfadj. real eigenval, ${\lambda }_i={\lambda }_i^{\ast }$
• $\sqrt{2}w=x+iy$, $(w,\bar{w})$ ONS $\Rightarrow (x,y)ONB$ of $span(w,\bar{w})$, useless? 15.2.9
• $f$ normal $\iff$ the matrix has $\lambda$s on the diagonal and weird $2\times 2$ blocks, 15.2.10
• $f$ orthogonal $\iff$ weird matrix .11 $\iff$ $f$ is isom. $\iff ⟨f(v),f(w)⟩=⟨v,w⟩\iff ||f(v)||=||v||$ .6

16

16.1 Teilbarkeit

• $a|b\iff {\exists }_{c}a\cdot c=b$, $a\widehat{=}b\iff a|b\wedge b|a\iff (a)=(b)$.
• We then define $\hat{a}⪯\hat{b}\iff a|b\iff (a)\subseteq (b)$, $ggT$ and $kgV$ defined.
• $a$ is a $gT$ of $B$ $\iff B\subseteq (a)$. $a$ is $ggT(B)$ $\Leftarrow (a)=(B)$ if $B$ is Hauptid. then $\Rightarrow$.

16.2 ID and PID

• in ID: $(ac=ab\wedge c\ne 0)\to a=b$, $ab=0\to (a=0\vee b=0)$
• in a PID there is always a $ggT$ and a $kgV$ same holds in a FD (.4.23)
• in an ID: $a\widehat{=}b\iff {\exists }_{x\in A^{\times }}a=bc\iff (a)=(b)$ every princ. ideal is unique up to mult. with a unity.

16.3 ggT

• $ggT(a,b)$ is the minimal ideal $(a,b)=(a,a-b)$ and also compute $mod.$ to simplify
• $ggT(a,b,c)$ where compute $d:=ggT(b,c)$ is easy, you can then just check if $a|d$
• in $\mathbb{Z}$ at least you can write $gcd(a,b)={\lambda }_1\cdot a+{\lambda }_2\cdot b$ to do that use the matrix as in .3
• in $\mathbb{K}[X]$ but also in $\mathbb{Z}$ for $gcd(a,b)$ you can compute $a=c\cdot b+d$ and then $gcd(a,b)=gcd(a,d)$.

16.4 Factorial Rings

• Define $p$ is prime iff. $p|ab\to (p|a\vee p|b)$ and irr. if $p=ab\to a\in A^{\times }\vee b\in A^{\times }$.
• $0$ is prime $\iff A$ is ID; also in ID $p\ne 0$ and prime $\Rightarrow p$ is irr.
• Def $\alpha :{\mathbb{P}}_A\Rightarrow {\mathbb{N}}_0$ $supp(\alpha ):=\{p\in {\mathbb{P}}_A|\alpha (p)\ne 0\}$. ${\mathbb{P}}_A^{\alpha }:={\prod }_{p\in supp(\alpha )}{p}^{\alpha (p)}$, $fac(a)=(c,\alpha )$ s.t. $a=c{\mathbb{P}}_A^{\alpha }$
• in any ID the factorisation $(c,\alpha )$ is unique
• in an FD every $a\in A\setminus \{0\}$ has a $(c,\alpha )\in A^{\times }\times {\mathbb{N}}_0$.
• in a FD $c{\mathbb{P}}_A^{\alpha }|d{\mathbb{P}}_A^{\beta }\iff \alpha ⪯\beta$; PID $\Rightarrow$ FD
• $A$ FD $\iff$ irr. $\to$ prime $\wedge$ for every $(a_n{)}_{n\in \mathbb{N}}$ s.t. $(a_1)\subseteq (a_2)\subseteq ...$ then $k\in \mathbb{N}$ s.t. $(a_k)=(a_{k+1})=...$

17

17.1 Smith

• Define Spalten- Zeilenoperationen, add and sub. and mult. it by $\lambda \in A^{\times }$.
• $Z\cdot A\cdot S=B$ for inv. mat. $Z$ and $S$, $A\sim B$, $|det(A)|=|det(B)|$, $|det(Z)|=|det(S)|=1$.
• We get a new $\sim$ and create new equivalence classes, rappresentative is Smith
• Smith: . ${\lambda }_1|...|{\lambda }_l$ for $l=\min (m,n)$ (16.4.17), it always exists.
• 17.3.5

17.2 Cauchy-Binet

• $A_{I,J}$ you forget useless rows and columns, it must not be square.
• $det(AB)={\sum }_{J\subseteq \{1,...,n\},|J|=m}(det(A_{I,J})(det(B_{J,I}))$ for $AB\in {\mathbb{K}}^{n\times n}$; $m>n\Rightarrow det(AB)=0$.
• $m>n\iff det(AB)=0$, $m=n\iff det(A)det(B)$.
• Minoref.: for $|L|=|I|$, $det((AB{)}_{I,L})={\sum }_{J\subseteq \{1,...,n\},|J|=m}(det(A_{I,J})(det(B_{J,L}))$, $AB$ mustn’t be $n\times n$
• minor${}_k$, the ideal of every minor of order $k$, ideal are decreasing for "$\subseteq$" .13

17.3

• $d_k(M)$ in the $gcd$ of all minors of order $k$, also $d_1(M)|...|d_k(M)$, (Determinantenteiler) .3
• $d_k(M)=c_1,...,c_k$, long way to get smith($M$), .4
• $c_j(M)$ is the $jj$th coeff. of $Smith(M)$; (Elementarteiler).
• $M\sim N\iff d(M)=d(N)$ .7.b, ez criterion for checking $\sim$.

17.4 Cayley-Hamilton

• $A\approx B\iff B=P^{-1}AP$ where $P$ is a change of basis $\Rightarrow$ they share everything
• $P_{i}A=A{P}_i\Rightarrow (PQ{)}_A=P_A{Q}_A$
• ==17.4.8 $P_A=0\iff {\exists }_{Q\in K[X{]}^{n\times n}}P=(X{I}_n-A)Q$==
• ==$P\cdot ch(b)=ch(A)\cdot Q\Rightarrow A{P}_A=P_{A}B$, 17.4.9==
• $A\approx B\iff ch(A)\sim ch(B)\iff d(ch(A))=d(ch(B))\iff$ same Smith.
• ext: $A\approx B\iff B=P^{-1}AP$ where $P$ is a change of basis $\Rightarrow$ they share everything
• Algorithm .12
  1. Compute $Smith(ch(A)|I_n)$ which gives you $(S|L)$
  2. Compute $Smith(ch(B))=:T$ note the Zeilenoperationen.
  3. Now take $L$ and inverte the Zeilenoperationen noted before this gives you $P$
  4. Decompose $P\in K[X{]}^{n\times n}$ in $X^{n-1}{P}_{n-1}+...+P_0$, for $P_{n-1},...,P_0\in K^{n\times n}$
  5. Now evaluate $B$ in $P$ to get $R:=P_B$ and define $R^{-1}BR=A$.
• $A{\approx }_{K}B\iff A{\approx }_{L}B$, for $K\subset L$ fields .14

17.5 Frobenius

• $Smith(ch(C_p))$ is easy $I_n$ con $p$ in fondo. $Frob(A)$ is like smith but with $C(p_1),...,C(p_m)$.
• Also $Smith(Ch(Frob(A)))$ is $I$ with $p_1,...,p_m$ on the end of the diagonal; $A\approx Frob(A)$.
• $c(ch(A))=(1,...,1,p_1,...,p_m)$ then $c_m={\mu }_A$ and ${\chi }_A=\prod c$.
• How to find$Frob(A)$: compute $Smith(ch(A))$, then read $p_1,...,p_m$.
• For any quad. $A$, there’s $(p_1,...,p_n),p_1|...|p_n$ with Frob(A), ${\mu }_A=p_m,{\chi }_A=(-1{)}^n{p}_1\cdot ...\cdot p_m$.
• To find $(p_1,...,p_m)$, compute $Smith(ch(A))$, there you have $p_1,...,p_m$.

17.6 Weierstraß

• $Wei(A)$ eind. bis auf Reih. and $p_1,...,p_m$ are powers of prime polyn. in $K[X]$; $A\approx Wei(A)$.
• How to compute: compute $Frob(A)$ then for every block ($>1\times 1$) split it in prime polyn .3.b

17.7 Jordan

• $J(\lambda ,n)$, ${\chi }_{J(\lambda ,n)}={\mu }_{J(\lambda ,n)}=(x-\lambda {)}^n$. For any Eigenwert you build a block respecting the mult.
• If we are in no ACF and ${\chi }_A$ doesn’t zerfällt, then the $Jord(A)$ is not defined.

Arithmetic Tricks

• $xy=\frac{1}{4}((x+y{)}^2-(x-y{)}^2)$, for Sylv. Sign and Cholesky
• $x^d-y^d=(x-y)(x^{d-1}+x^{d-2}y+...+y^{d-1})$
• $(AB{)}^T=B^T{A}^T$ 11.3.5
• for $V,W$ $F$-VS, $|V|=|W|>|F|\Rightarrow V\cong W$
• $deg({\chi }_A)=n$ for $A\in K^{n\times n}$
• $I\cap J$ is an ideal

To Memorise

• Gram-Schm. want $(w_i{)}_{i\in I}$ $w_1=v_1$, $w_2=v_2-\frac{⟨v_2,w_1⟩}{|w_1^2|}{w}_1$, $w_3=v_3-\frac{⟨v_3,w_1⟩}{|w_1{|}^2}{w}_1-\frac{⟨v_3,w_2⟩}{|w_2{|}^2}{w}_2$.
• $det(AB)={\sum }_{J\subseteq \{1,...,n\},|J|=m}(det(A_{I,J})(det(B_{J,I}))$ for $AB\in {\mathbb{K}}^{n\times n}$; $m>n\Rightarrow det(AB)=0$.
• Minoref.: for $|L|=|I|$, $det((AB{)}_{I,L})={\sum }_{J\subseteq \{1,...,n\},|J|=m}(det(A_{I,J})(det(B_{J,L}))$, $AB$ mustn’t be $n\times n$