Example: Cholesky Factorization of Real Matrices

Functions > Vector and Matrix > Matrix Factorization > Example: Cholesky Factorization of Real Matrices

Use the Cholesky function to perform Cholesky factorization of a real matrix.

To avoid logical mismatches when performing boolean comparisons, enable Approximate Equality in the Calculation Options drop-down list.

1. Define a real positive definite square matrix M.

2. Apply the eigenvals function to ensure that the matrix is positive definite.

3. Apply the rank function to ensure that M is a full-rank matrix.

4. Set arguments p and u to control the enabling/disabling of pivoting and lower/upper factorization.

5. Use the Cholesky function to perform the default factorization of matrix M - with pivoting and lower factorization.

The default function Cholesky(M) is equivalent to Cholesky(M,1,0).

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 <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">Cholesky</id>
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 <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">Cholesky</id>
 <sequence>
 <id xml:space="preserve">M</id>
 <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">
 p<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
 </id>
 <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">
 u<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript>
 </id>
 </sequence>
 </apply>
 </apply>
 <unitOverride>
 <placeholder />
 </unitOverride>
 </eval>
 </math>
</region>

6. Show that P10T x M x P10 = L10 x L10T.

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<region id="ID0ESQ1EB" actualWidth="202.72666666666666" actualHeight="62.800000000000004" top="1094.4" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="86"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <apply> <mult /> <apply> <transpose /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P10</id> </apply> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M</id> </apply> <id xml:space="preserve">P10</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>	<region id="ID0EAR1EB" actualWidth="177.81666666666666" actualHeight="62.800000000000004" top="1094.4" left="307.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="88"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">L10</id> <apply> <transpose /> <id xml:space="preserve">L10</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>
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The relationship is logically true.

7. Use the Cholesky function to perform factorization of matrix M - with no pivoting and lower factorization (default).

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 <define xmlns="http://schemas.mathsoft.com/math50">
 <id labels="VARIABLE" xml:space="preserve">m00</id>
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 <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">Cholesky</id>
 <sequence>
 <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M</id>
 <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">
 p<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript>
 </id>
 </sequence>
 </apply>
 <unitOverride>
 <placeholder />
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 </define>
 <resultFormat>
 <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="true" />
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Not specifying argument u, as in Cholesky(M, 0), is equivalent to setting it to 0 as in Cholesky(M, 0, 0).

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 p<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript>
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 <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">Cholesky</id>
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 <id xml:space="preserve">M</id>
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 p<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript>
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 <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">
 u<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript>
 </id>
 </sequence>
 </apply>
 </apply>
 <unitOverride>
 <placeholder />
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 <resultFormat>
 <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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The returned lower matrix, L10, when pivoting is enabled is NOT equal to the returned lower matrix, L00, when pivoting is disabled.

The relationship is logically false.

8. Show that M = L00 x L00T.

The relationship is logically true.

9. Use the Cholesky function to perform factorization of matrix M - with pivoting and upper factorization.

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 p<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
 </id>
 <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">
 u<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
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 <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="true" />
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10. Show that P11T x M x P11 = U11T x U11.

The relationship is logically true.

11. Use the Cholesky function to perform factorization of matrix M - with no pivoting and upper factorization.

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 p<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript>
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 u<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
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12. Show that M = U01T x U01.

The relationship is logically true.