Example: Cholesky Factorization of Complex Matrices
Use the Cholesky function to perform Cholesky factorization of a complex Hermitian matrix.
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To avoid logical mismatches when performing boolean comparisons, enable Approximate Equality in the Calculation Options drop-down list.
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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) |
5. Show that P10T x M x P10 = L10 x conj(L10T).
The relationship is logically true.
6. Use the Cholesky function to perform factorization of matrix M - with no pivoting and lower factorization (default).
| Not specifying argument u, as in Cholesky(M, 0), is equivalent to setting it to 0 as in Cholesky(M, 0, 0). |
| 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.
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.
11. Show that M = conj(U01T) x U01.
The relationship is logically true.
