E
EmmanuelMiserandini
Guest
Unicity condition of L requires a supplementary condition on L : the diagonal must be strictly positive. I added that in the statement
[td][/td] [td]== Statement ==[/td]
[td]== Statement ==[/td] [td]The Cholesky decomposition of a [[Hermitian matrix|Hermitian]] [[positive-definite matrix]] {{math|'''A'''}}, is a decomposition of the form[/td]
[td]The Cholesky decomposition of a [[Hermitian matrix|Hermitian]] [[positive-definite matrix]] {{math|'''A'''}} is a decomposition of the form[/td] [td][/td]
[td][/td] [td]<math display=block>\mathbf{A} = \mathbf{L L}^{*},</math>[/td]
[td]<math display=block>\mathbf{A} = \mathbf{L L}^{*},</math>[/td] [td][/td]
[td][/td] [td]where {{math|'''L'''}} is a [[lower triangular matrix]] with real and positive diagonal entries, and {{math|'''L'''}}* denotes the [[conjugate transpose]] of {{math|'''L'''}}. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition.<ref>{{harvtxt|Golub|Van Loan|1996|p=143}}, {{harvtxt|Horn|Johnson|1985|p=407}}, {{harvtxt|Trefethen|Bau|1997|p=174}}.</ref>[/td]
[td]where {{math|'''L'''}} is a [[lower triangular matrix]] with real and positive diagonal entries, and {{math|'''L'''}}* denotes the [[conjugate transpose]] of {{math|'''L'''}}. Every Hermitian positive-definite matrix (and thus also every real symmetric positive-definite matrix) has a Cholesky decomposition and the lower triangular matrix is unique if we impose the diagonal to be strictly positive.<ref>{{harvtxt|Golub|Van Loan|1996|p=143}}, {{harvtxt|Horn|Johnson|1985|p=407}}, {{harvtxt|Trefethen|Bau|1997|p=174}}.</ref>[/td] [td][/td]
[td][/td] [td]The converse holds trivially: if {{math|'''A'''}} can be written as {{math|'''LL'''*}} for some invertible {{math|'''L'''}}, lower triangular or otherwise, then {{math|'''A'''}} is Hermitian and positive definite.[/td]
[td]The converse holds trivially: if {{math|'''A'''}} can be written as {{math|'''LL'''*}} for some invertible {{math|'''L'''}}, lower triangular or otherwise, then {{math|'''A'''}} is Hermitian and positive definite.[/td]
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[td][/td]Revision as of 07:26, 4 September 2025
[/td][td][/td] [td]== Statement ==[/td]
[td]== Statement ==[/td] [td]The Cholesky decomposition of a [[Hermitian matrix|Hermitian]] [[positive-definite matrix]] {{math|'''A'''}}, is a decomposition of the form[/td]
[td]The Cholesky decomposition of a [[Hermitian matrix|Hermitian]] [[positive-definite matrix]] {{math|'''A'''}} is a decomposition of the form[/td] [td][/td]
[td][/td] [td]<math display=block>\mathbf{A} = \mathbf{L L}^{*},</math>[/td]
[td]<math display=block>\mathbf{A} = \mathbf{L L}^{*},</math>[/td] [td][/td]
[td][/td] [td]where {{math|'''L'''}} is a [[lower triangular matrix]] with real and positive diagonal entries, and {{math|'''L'''}}* denotes the [[conjugate transpose]] of {{math|'''L'''}}. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition.<ref>{{harvtxt|Golub|Van Loan|1996|p=143}}, {{harvtxt|Horn|Johnson|1985|p=407}}, {{harvtxt|Trefethen|Bau|1997|p=174}}.</ref>[/td]
[td]where {{math|'''L'''}} is a [[lower triangular matrix]] with real and positive diagonal entries, and {{math|'''L'''}}* denotes the [[conjugate transpose]] of {{math|'''L'''}}. Every Hermitian positive-definite matrix (and thus also every real symmetric positive-definite matrix) has a Cholesky decomposition and the lower triangular matrix is unique if we impose the diagonal to be strictly positive.<ref>{{harvtxt|Golub|Van Loan|1996|p=143}}, {{harvtxt|Horn|Johnson|1985|p=407}}, {{harvtxt|Trefethen|Bau|1997|p=174}}.</ref>[/td] [td][/td]
[td][/td] [td]The converse holds trivially: if {{math|'''A'''}} can be written as {{math|'''LL'''*}} for some invertible {{math|'''L'''}}, lower triangular or otherwise, then {{math|'''A'''}} is Hermitian and positive definite.[/td]
[td]The converse holds trivially: if {{math|'''A'''}} can be written as {{math|'''LL'''*}} for some invertible {{math|'''L'''}}, lower triangular or otherwise, then {{math|'''A'''}} is Hermitian and positive definite.[/td]
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