Example: Cholesky Factorization of Complex Matrices

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

Use the Cholesky function to perform Cholesky factorization of a complex Hermitian matrix.

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

1. Define a complex Hermitian definite square matrix M.

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

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

4. 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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 p<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</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>
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5. Show that P10T x M x P10 = L10 x conj(L10T).

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The relationship is logically true.

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

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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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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.

7. Show that M = L00 x conj(L00T).

The relationship is logically true.

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

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9. Show that P11T x M x P11 = conj(U11T) x U11.

The relationship is logically true.

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

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11. Show that M = conj(U01T) x U01.

The relationship is logically true.