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 čemer velja:
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{\displaystyle (f(x))^{j}=\sum _{k=0}^{\infty }M[f]_{jk}x^{k}\!\,.}
Tako se lahko zapiše 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 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},\ldots \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 se določi tudi ničelno potenco funkcije
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{\displaystyle f(x)\!\,}
M
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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},\ldots \right]^{\tau }}
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{\displaystyle M[f]*\left[1,x,x^{2},x^{3},\ldots \right]^{\tau }=\left[1,f(x),(f(x))^{2},(f(x))^{3},\ldots \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 čemer 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
f
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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
M
{\displaystyle M\!\,}
f
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x
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{\displaystyle f(x)\!\,}
B
{\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 značilnosti:
M
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{\displaystyle \,M[f^{n}]=M[f]^{n}\!\,,}
kjer je:
f
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{\displaystyle \,f^{n}\!\,}
iteracija funkcije
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{\displaystyle \,M[f^{-1}]=M[f]^{-1}\!\,,}
kjer je:
f
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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/445bab7acc7582ca1c09dab1b9bb97e3eb221aaf)
![{\displaystyle (f(x))^{j}=\sum _{k=0}^{\infty }M[f]_{jk}x^{k}\!\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/404dcdb6654f87a198005a43296c8e8b010fa053)
![{\displaystyle f(x)=\sum _{k=0}^{\infty }M[f]_{1,k}x^{k}\!\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae2750e62f1ce201b506d2b8b20a927427d1859)
![{\displaystyle M[f]\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/218f554334c089bda04dfa20416770295c1b731f)
![{\displaystyle \left[1,x,x^{2},x^{3},\ldots \right]^{\tau }\!\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0600f4232d3abd7886dc481a6d87ff4bde28765)
![{\displaystyle f(x)^{2}=\sum _{k=0}^{\infty }M[f]_{2,k}x^{k}\!\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2fa584021c97b3a4ea802923f663c882cbb3ded)
![{\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/2b3199c8877cdc696572f23e23690df8c1e6f376)
![{\displaystyle \left[1,x,x^{2},x^{3},\ldots \right]^{\tau }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6450736a896d11784ac1572f0618f2747b1756)
![{\displaystyle M[f]*\left[1,x,x^{2},x^{3},\ldots \right]^{\tau }=\left[1,f(x),(f(x))^{2},(f(x))^{3},\ldots \right]^{\tau }\!\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c119ecdd34f008f97f1f4163752abea17bc5e8d)
![{\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/ec6f270ea0c3358be93b0576b9c17b208cdd228e)
![{\displaystyle (f(x))^{k}=\sum _{j=0}^{\infty }B[f]_{jk}x^{j}\!\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e2780a61988a2e27745a3b17bd7105db193a09)
![{\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/732bfe646adf96a09001be96ebf0d122326384eb)
![{\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/e9a1072db922a35f9376d99723725675df5d870b)
![{\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/2c784d4f0b63cfd8acb6abf187cf9503f74e0769)
![{\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/b5f6eba28566fd98b3e6b2216e63b62f147219d4)
![{\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/b9307721086e18250d806c99d650376b3db77cf7)
![{\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/fc859bf3a6aa7e740d8793143f602b0ffb91cdfd)
![{\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/76195a62698eb125477b25e9a7bdf719b73b032c)
![{\displaystyle M[f\circ g]=M[f]M[g]\!\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47a8e44e05fe7f7523bab785ba0863c8bc7b1c23)
![{\displaystyle B[f\circ g]=B[g]B[f]\!\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f36def24c4f3d8d3f8a505e5ec0a5d0ef4b1b5)
![{\displaystyle \,M[f^{n}]=M[f]^{n}\!\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7502899575e2869d6eddb22dcd3b69ec2f5c595)
![{\displaystyle \,M[f^{-1}]=M[f]^{-1}\!\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72d5c2cbc2a30e150430cfeb1dad636d5d4a797e)
