Carlemanova matrika je matrika , ki se uporablja za pretvorbo kompozituma funkcij v množenje matrik . Druga področja uporabe so še v teoriji iteracij in verjetnosti ter v Markovskih verigah .
Calermanova matrika funkcije
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{\displaystyle f(x)\,}
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{\displaystyle M[f]_{jk}={\frac {1}{k!}}\left[{\frac {d^{k}}{dx^{k}}}(f(x))^{j}\right]_{x=0}}
pri tem pa velja
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{\displaystyle (f(x))^{j}=\sum _{k=0}^{\infty }M[f]_{jk}x^{k}.}
Tako lahko zapišemo določanje funkcije
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{\displaystyle f(x)\,}
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{\displaystyle f(x)=\sum _{k=0}^{\infty }M[f]_{1,k}x^{k}.}
kar pa je skalarni produkt prve vrstice matrike
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{\displaystyle M[f]}
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{\displaystyle \left[1,x,x^{2},x^{3},...\right]^{\tau }}
Množenje z drugo vrstico matrike
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{\displaystyle M[f]}
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{\displaystyle f(x)\,}
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{\displaystyle f(x)^{2}=\sum _{k=0}^{\infty }M[f]_{2,k}x^{k}.}
Lahko pa določimo tudi ničelno potenco funkcije
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{\displaystyle f(x)\,}
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{\displaystyle M[f]}
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{\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
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{\displaystyle M[f]}
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{\displaystyle \left[1,x,x^{2},x^{3},...\right]^{\tau }}
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{\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 funkcije
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{\displaystyle f(x)\,}
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{\displaystyle B[f]_{jk}={\frac {1}{j!}}\left[{\frac {d^{j}}{dx^{j}}}(f(x))^{k}\right]_{x=0}}
pri tem pa velja
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{\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:
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{\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:
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{\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:
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{\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:
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{\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:
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{\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
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{\displaystyle f(x)=\sum _{k=1}^{\infty }f_{k}x^{k}}
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{\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
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{\displaystyle f(x)=\sum _{k=0}^{\infty }f_{k}x^{k}}
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{\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)}
Obe matriki (Carlemanova in Bellova) zadoščata osnovnima odnosoma:
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{\displaystyle M[f\circ g]=M[f]M[g]}
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{\displaystyle B[f\circ g]=B[g]B[f]}
kar dela Carlemanovo matriko
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{\displaystyle M\,}
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{\displaystyle f(x)\,}
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{\displaystyle B\,}
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{\displaystyle f(x)}
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{\displaystyle f\circ g}
kompozitum funkcij
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{\displaystyle f(g(x))}
Razen tega veljata še naslednji lastnosti:
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{\displaystyle \,M[f^{n}]=M[f]^{n}}
kjer je
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{\displaystyle \,f^{n}}
iteracija funkcije
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{\displaystyle \,M[f^{-1}]=M[f]^{-1}}
kjer je
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{\displaystyle \,f^{-1}}
inverzna funkcija (če je Calermanova matrika obrnljiva).
![{\displaystyle M[f]_{jk}={\frac {1}{k!}}\left[{\frac {d^{k}}{dx^{k}}}(f(x))^{j}\right]_{x=0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d72f42ebd02bfcc2b5e596df6bbc44b9a580f8a2)
![{\displaystyle (f(x))^{j}=\sum _{k=0}^{\infty }M[f]_{jk}x^{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03463b6cd637cee2f67b82f27d2090ea727f8911)
![{\displaystyle f(x)=\sum _{k=0}^{\infty }M[f]_{1,k}x^{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97807ce89de4cb75145c9d6d1b0d6cc28cc01cb2)
![{\displaystyle M[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9f1b98309f90e9a854df2a6d8ab0fc2829b362)
![{\displaystyle \left[1,x,x^{2},x^{3},...\right]^{\tau }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b1e11cc42c4a83a36d634fbdfffd5a91bd3de3)
![{\displaystyle f(x)^{2}=\sum _{k=0}^{\infty }M[f]_{2,k}x^{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7087aa822372ac806df2816e6b998e853b4d546a)
![{\displaystyle f(x)^{0}=1=\sum _{k=0}^{\infty }M[f]_{0,k}x^{k}=1+\sum _{k=1}^{\infty }0*x^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02b32a42b727d226eeec85f9aaf8953d706bb5a9)
![{\displaystyle M[f]*\left[1,x,x^{2},x^{3},...\right]^{\tau }=\left[1,f(x),(f(x))^{2},(f(x))^{3},...\right]^{\tau }\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9796e13a363ea6c69a42f900930c8c754a3a19f)
![{\displaystyle B[f]_{jk}={\frac {1}{j!}}\left[{\frac {d^{j}}{dx^{j}}}(f(x))^{k}\right]_{x=0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/430d5972a569a725d01a79b918fdbdab7d31857b)
![{\displaystyle (f(x))^{k}=\sum _{j=0}^{\infty }B[f]_{jk}x^{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/330b82e2a3e05a007b8ec2650fcd2f42a1bf5076)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27037f56eb81c02bca3637d7fa1a64a7acf69290)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/060db1559fd634af4732397f145102b847ee28d0)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3fea6f7f68d36e2bd565f790c580d6cf3638c7)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d29dc047a6112f5ff455b1dffd13a54b90102b18)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3518b703f7d0701200e12ef02d74528bb03450)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa0961a60884bc09a9405dea90f500a8747aea25)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8287cbf2b6aa4ac5e2488a78832425dae8c6ae)
![{\displaystyle M[f\circ g]=M[f]M[g]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9e32e0d92b2de0c71b4f34f6633ecab7beb603)
![{\displaystyle B[f\circ g]=B[g]B[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52e5ea84b994367981d977ca125a3941f3a3f4d2)
![{\displaystyle \,M[f^{n}]=M[f]^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91cc76a4e0e2ad77fa8fa75e53df442147103b07)
![{\displaystyle \,M[f^{-1}]=M[f]^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd8b441bfa27dc96eca74c5ebe051d83a5bbe6c)
