Definicija
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Calermanova matrika funkcije f ( x ) {\displaystyle f(x)\,} je določena z
M [ f ] j k = 1 k ! [ d k d x k ( f ( x ) ) j ] x = 0 {\displaystyle M[f]_{jk}={\frac {1}{k!}}\left[{\frac {d^{k}}{dx^{k}}}(f(x))^{j}\right]_{x=0}} pri tem pa velja
( f ( x ) ) j = ∑ k = 0 ∞ M [ f ] j k x k . {\displaystyle (f(x))^{j}=\sum _{k=0}^{\infty }M[f]_{jk}x^{k}.} Tako lahko zapišemo določanje funkcije f ( x ) {\displaystyle f(x)\,} kot
f ( x ) = ∑ k = 0 ∞ M [ f ] 1 , k x k . {\displaystyle f(x)=\sum _{k=0}^{\infty }M[f]_{1,k}x^{k}.} ,kar pa je skalarni produkt prve vrstice matrike M [ f ] {\displaystyle M[f]} z
vektorjem [ 1 , x , x 2 , x 3 , . . . ] τ {\displaystyle \left[1,x,x^{2},x^{3},...\right]^{\tau }} .
Množenje z drugo vrstico matrike M [ f ] {\displaystyle M[f]} nam da drugo potenco funkcije f ( x ) {\displaystyle f(x)\,}
f ( x ) 2 = ∑ k = 0 ∞ M [ f ] 2 , k x k . {\displaystyle f(x)^{2}=\sum _{k=0}^{\infty }M[f]_{2,k}x^{k}.} .Lahko pa določimo tudi ničelno potenco funkcije f ( x ) {\displaystyle f(x)\,} . V matriki M [ f ] {\displaystyle M[f]} predpostavimo, da vrstica 0 vsebuje ničle povsod, razen na prvem mestu. To nam da
f ( x ) 0 = 1 = ∑ k = 0 ∞ M [ f ] 0 , k x k = 1 + ∑ k = 1 ∞ 0 ∗ x k {\displaystyle f(x)^{0}=1=\sum _{k=0}^{\infty }M[f]_{0,k}x^{k}=1+\sum _{k=1}^{\infty }0*x^{k}} Skalarni produkt matrike M [ f ] {\displaystyle M[f]} z vektorjem [ 1 , x , x 2 , x 3 , . . . ] τ {\displaystyle \left[1,x,x^{2},x^{3},...\right]^{\tau }} nam da vektor
M [ f ] ∗ [ 1 , x , x 2 , x 3 , . . . ] τ = [ 1 , f ( x ) , ( f ( x ) ) 2 , ( f ( x ) ) 3 , . . . ] τ {\displaystyle M[f]*\left[1,x,x^{2},x^{3},...\right]^{\tau }=\left[1,f(x),(f(x))^{2},(f(x))^{3},...\right]^{\tau }\,} . Bellova matrika
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Bellova matrika funkcije f ( x ) {\displaystyle f(x)\,} je določena kot
B [ f ] j k = 1 j ! [ d j d x j ( f ( x ) ) k ] x = 0 {\displaystyle B[f]_{jk}={\frac {1}{j!}}\left[{\frac {d^{j}}{dx^{j}}}(f(x))^{k}\right]_{x=0}} pri tem pa velja
( f ( x ) ) k = ∑ j = 0 ∞ B [ f ] j k x j {\displaystyle (f(x))^{k}=\sum _{j=0}^{\infty }B[f]_{jk}x^{j}} To pa pomeni, da je Bellova matrika transponirana Carlemanova matrika.
Carlemanova matrika konstante je:
M [ a ] = ( 1 0 0 ⋯ a 0 0 ⋯ a 2 0 0 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M[a]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&0&0&\cdots \\a^{2}&0&0&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} Carlemanova matrika identične funkcije je:
M x [ x ] = ( 1 0 0 ⋯ 0 1 0 ⋯ 0 0 1 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[x]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&1&0&\cdots \\0&0&1&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} Carlemanova matrika dodane konstante je:
M x [ a + x ] = ( 1 0 0 ⋯ a 1 0 ⋯ a 2 2 a 1 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[a+x]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&1&0&\cdots \\a^{2}&2a&1&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} Carlemanova matrika zmnožka s konstanto je:
M x [ c x ] = ( 1 0 0 ⋯ 0 c 0 ⋯ 0 0 c 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[cx]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&c&0&\cdots \\0&0&c^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} Carlemanova matrika linearne funkcije je:
M x [ a + c x ] = ( 1 0 0 ⋯ a c 0 ⋯ a 2 2 a c c 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M_{x}[a+cx]=\left({\begin{array}{cccc}1&0&0&\cdots \\a&c&0&\cdots \\a^{2}&2ac&c^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} Carlemanova matrika funkcije f ( x ) = ∑ k = 1 ∞ f k x k {\displaystyle f(x)=\sum _{k=1}^{\infty }f_{k}x^{k}} je:
M [ f ] = ( 1 0 0 ⋯ 0 f 1 f 2 ⋯ 0 0 f 1 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M[f]=\left({\begin{array}{cccc}1&0&0&\cdots \\0&f_{1}&f_{2}&\cdots \\0&0&f_{1}^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} Carlemanova matrika funkcije f ( x ) = ∑ k = 0 ∞ f k x k {\displaystyle f(x)=\sum _{k=0}^{\infty }f_{k}x^{k}} je:
M [ f ] = ( 1 0 0 ⋯ f 0 f 1 f 2 ⋯ f 0 2 2 f 0 f 1 f 1 2 ⋯ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle M[f]=\left({\begin{array}{cccc}1&0&0&\cdots \\f_{0}&f_{1}&f_{2}&\cdots \\f_{0}^{2}&2f_{0}f_{1}&f_{1}^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{array}}\right)} . Lastnosti matrike
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