Added Orthonormal basis
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alexander
@@ -443,59 +443,10 @@
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content((0.5, 0.5), $A$)
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})
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)
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#table(
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columns: (auto, 1fr),
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inset: 2mm,
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fill: (x, y) => if (calc.rem(y, 2) == 0) { tableFillLow } else { tableFillHigh },
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[*Einheits Matrix*\ $I,E$], [],
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[*Diagonalmatrix* \ $Sigma,D$], [
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Nur Einträger auf Hauptdiagonalen \
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$det(D) = d_00 dot d_11 dot d_22 dot ...$
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],
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[*Symetrisch*\ $S$], [
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$S = S^T$, $S in KK^(n times n)$\
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$A A^T$, $A^T A$ ist symetrisch \
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$S$ immer diagonaliserbar \
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EW immer $in RR$, EV orthogonal
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],
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[*Invertierbar*], [
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$exists A^(-1) : A A^(-1) = A^(-1) A = E$ \
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*Invertierbar wenn:* \
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$A$ bijektiv, $det(A) = 0$ \
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$"Spalten Vekoren lin. unabhänig"$ \
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$Rang(A) = n, A in KK^(n times n)$ \
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$det(A) = 0$ \
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*Nicht Invertierbar wenn:*\
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$exists$ EW $!= 0 => not "invertierbar"$
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Keine Qudratische Matrix
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],
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[*Orthogonal*\ $O$], [
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$O^T = O^(-1)$ \
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$ip(O v, O w) = ip(v, w)$
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],
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[*Unitair*], [
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$V^* )$
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],
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[*Diagonaliserbar*], [
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$exists A = B D B^(-1)$, $D$ diagonal,
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$B$: Splaten sind EV von $A$
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- Selbst-Adujunkte diagonalisierbar
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- Symetrisch Matrix
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- $A in KK^(n times n) "AND" alg(lambda) = geo(lambda)$
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],
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[*postiv-semi-definit*], [
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$forall$ EW $>= 0$
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],
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)
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]
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#bgBlock(fill: colorAbbildungen)[
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#subHeading(fill: colorAbbildungen)[Determinate und Bilinearform]
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#subHeading(fill: colorAbbildungen)[Linearform]
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- Sclar-Produkt $ip(ve(a), ve(b))$ ist Bi-Linearform
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- Symetrisch
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@@ -515,14 +466,28 @@
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- $ve(v)_1, ... "linear abhänig" <=> f(ve(v)_1, ...) = 0$
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- $ve(v)_1, ... "linear unabhänig" <=> f(ve(v)_1, ...) != 0$, eindeutig
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#SeperatorLine
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- *Positiv definite symetrisch Bilinearform*
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- $equiv$ Skalarprodukt $ip(dot, dot)$ in $RR$
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- $f(ve(v)_1, ve(v)_2) = f(ve(v)_2, ve(v)_1), space space forall ve(v)_1, ve(v)_2 in KK^n$
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- linear in beiden Argumenten
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- $f(ve(v), ve(v)) > 0, v in V without {ve(0)}$
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]
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#bgBlock(fill: colorAbbildungen)[
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#subHeading(fill: colorAbbildungen)[Determinante]
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*Determinaten Form* \
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Nicht tiviale ($f(...) = 0$) n-Linearform auf einem VR.
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Nicht tiviale ($f(...) = 0$) alternierende n-Linearform auf einem VR.
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$exists$ Immer, in jeder Scalierung
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Speziell für Martizen $in KK^(n times n)$ \ (Qudratische, Endomorphismus)
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*Herleitung:*
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Für die $det equiv delta$: $delta(e_1, e_2, e_3, ...) = 1$, alternierend und n-linearform
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1. Mit Linearität zerlegen
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2. Mit Alterniered, Element tauschen: $delta(e_1, e_2, e_3, ...) dot ...$
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*Leibniz-Formel*
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$det(A) = limits(sum)_(sigma in S_n) sign(sigma)( a_(sigma(1)1) dot a_(sigma(2)2) dot dots dot a_(sigma(n)n))$
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@@ -611,11 +576,50 @@
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]
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#bgBlock(fill: colorVR)[
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#subHeading(fill: colorVR)[Eukldische Vektorräume]
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]
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#subHeading(fill: colorVR)[Eukldische/Unitair Vektorräume]
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*Kannonische Scalar Produkt*
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$ip(ve(x), ve(y)) := limits(sum)_(i=1)^n x_i y_i$
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- Positiv definit: $ip(ve(x), ve(x)) > 0$
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- Linear in beiden Argument: \
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$ip(lambda ve(x), ve(y)) = lambda ip(ve(x), ve(y))$\
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$ip(ve(x) + ve(a), ve(y)) = ip(ve(x), ve(y)) + ip(ve(a), ve(y))$
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*Norm*
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- $norm(ve(v)) = 0 <=> ve(v) = ve(0)$
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- $norm(lambda ve(v)) = abs(lambda) norm(ve(v))$
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- Dreieckesungleichung: $norm(x + y) <= norm(x) + norm(y)$
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Generisch/$L_p$-Norm: $|| v ||_p = root(p, sum_(k=1)^n (x_k)^p)$
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*Induzierte Norm:* $norm(ve(v)) = sqrt(ip(ve(v), ve(v)))$ (Bliebiges $ip(dot, dot)$)
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*Eukldische Norm:*
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- $L_2$-Norm/kannoische Norm: $norm(ve(v)) = sqrt(ip(ve(v), ve(v)))$
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*Cauchy-Schwarz-Ungleichung:* $abs(ip(v, w)) <= norm(v) norm(w)$
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- Gilt in Eukldische Vektoraum
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- Gilt nur mit aus Eukldischer Norm
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*Euklidsche Vektorraum:*
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- $ = (RR^n"-Vekorraum", ip(dot, dot))$, (Irgendeine Skalarprodukt)
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- Eigenschaften:
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- Polarisation: $ip(v, w) = 1/4 (norm(v + w)^2 - norm(v -w )^2)$
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- Parallelogrammgleichung: \
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$2 (norm(v)^2 + norm(w)^2) = norm(v + w)^2 + norm(v - w)^2$
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- Winkel: $cos alpha = ip(v, w)/(norm(v) norm(w))$
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*Orthogonal Vektoren:* $ip(v, w) = 0$ \
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*Ortho#text(red)[normal] Vektoren:* $ip(v, w) = 0$ UND $norm(v),norm(w) = 1$
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#SeperatorLine
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#colbreak()
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*Orthogonal Projektion*
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$U subset V$ Untervektorraum eines Eukldische VRs $V$, $U$ orthogo#text(red)[normal]: $pi_U(v) = limits(sum)_(i=1)^k ip(v, u_i) u_i$
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#bgBlock(fill: colorVR)[
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#subHeading(fill: colorVR)[Unitair Vektorräume ]
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]
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#bgBlock(fill: colorMatrixVerfahren)[
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@@ -663,6 +667,14 @@
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#bgBlock(fill: colorMatrixVerfahren)[
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#subHeading(fill: colorMatrixVerfahren)[Gram-Schmit ONB]
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Idee: $ip("Orth"#text(red)[normal] v, x) = "Anteil von" x "an" v$ \
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Ziel: Basis $W -->$ Orthogonal Bais $V$
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1. $v_1 = 1/norm(w_1)$
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2. $hat(v)_(j+1) = w_(j+1) -ip(w_(j+1), v_1)v_1 - ip(w_(j+2), v_2)v_2 ... $
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3. $v_(j+1) = hat(v)_(j+1)/norm(hat(v)_(j+1))$
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]
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@@ -744,21 +756,18 @@
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$S = mat(sigma_0, 0; 0, sigma_1; dots.v, dots.v; 0, 0) quad quad quad S = mat(sigma_0, 0, dots, 0; 0, sigma_1, ..., 0)$
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]
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#colbreak()
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Matrix Normen]
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$|| dot ||_M$ Matrix Norm, $|| dot ||_V$ Vektornorm
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Generisch Vektor Norm: $|| v ||_p = root(p, sum_(k=1)^n (x_k)^p)$
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- submultiplikativ: $||A B||_"M" <= ||A||||B||$
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- verträglich mit einer Vektornorm: $||A v||_"V" <= ||A||_"M" ||v||_"V"$
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*Frobenius-Norm* $||A||_"M" = sqrt(sum_(i=1)^m sum_(j=1)^n a_(m n)^2)$
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*Induzierte Norm* \
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$||A||_"M" = sup_(v in V without {0}) (||A v||_V)/(||v||_V) = sup_(||v|| = 1) (||A v||_V)/(||v||_V)$
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$||A||_"M" = sup_(v in V without {0}) (||A v||_V)/(||v||_V) = sup_(||v|| = 1) ||A v||_V$
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- submultiplikativ
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- verträglich mit einer Vektornorm $||dot||_V$
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@@ -794,3 +803,65 @@
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#sinTable
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]
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#columns(2)[
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#table(
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columns: (auto, 1fr),
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inset: 2mm,
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fill: (x, y) => if (calc.rem(y, 2) == 0) { tableFillLow } else { tableFillHigh },
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[*Einheits Matrix*\ $I,E$], [
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$det(E) = 1$
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],
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[*Diagonalmatrix* \ $Sigma,D$], [
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Nur Einträger auf Hauptdiagonalen \
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$det(D) = d_00 dot d_11 dot d_22 dot ...$
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],
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[*Symetrisch*\ $S$], [
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$S = S^T$, $S in KK^(n times n)$\
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$A A^T$, $A^T A$ ist symetrisch \
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$S$ immer diagonaliserbar \
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EW immer $in RR$, EV orthogonal
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],
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[*Invertierbar*], [
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$exists A^(-1) : A A^(-1) = A^(-1) A = E$ \
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*Invertierbar wenn:* \
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$A$ bijektiv, $det(A) = 0$ \
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$"Spalten Vekoren lin. unabhänig"$ \
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$Rang(A) = n, A in KK^(n times n)$ \
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$det(A) = 0$ \
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*Nicht Invertierbar wenn:*\
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$exists$ EW $!= 0 => not "invertierbar"$
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Keine Qudratische Matrix
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],
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[*Orthogonal*], [
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#grid(
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columns: (1fr, 1fr),
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[
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- Immer Bijektiv
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- $det (A) = plus.minus 1$
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$O^T = O^(-1) quad quad O^T O = O O^T = I$ \
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$ip(O v, O w) = ip(v, w)$
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]
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)
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],
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[*Unitair* $equiv$ Orthogonal $in CC$], [
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- Immer Bijektiv
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$V = V^*$
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],
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[*Diagonaliserbar*], [
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$exists A = B D B^(-1)$, $D$ diagonal,
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$B$: Splaten sind EV von $A$
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- Selbst-Adujunkte ($$) diagonalisierbar
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- Symetrisch Matrix
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- $A in KK^(n times n) "UND" alg(lambda) = geo(lambda)$
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],
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[*postiv-semi-definit*], [
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$forall$ EW $>= 0$
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],
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)
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]
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