Proof For Positive Semi-definite Matrices
The above algorithms show that every positive definite matrix A has a Cholesky decomposition. This result can be extended to the positive semi-definite case by a limiting argument. The argument is not fully constructive, i.e., it gives no explicit numerical algorithms for computing Cholesky factors.
If A is an n-by-n positive semi-definite matrix, then the sequence {Ak} = {A + (1/k)In} consists of positive definite matrices. (This is an immediate consequence of, for example, the spectral mapping theorem for the polynomial functional calculus.) Also,
- Ak → A
in operator norm. From the positive definite case, each Ak has Cholesky decomposition Ak = LkL*k. By property of the operator norm,
So {Lk} is a bounded set in the Banach space of operators, therefore relatively compact (because the underlying vector space is finite dimensional). Consequently it has a convergent subsequence, also denoted by {Lk}, with limit L. It can be easily checked that this L has the desired properties, i.e. A = LL* and L is lower triangular with non-negative diagonal entries: for all x and y,
Therefore A = LL*. Because the underlying vector space is finite dimensional, all topologies on the space of operators are equivalent. So Lk tends to L in norm means Lk tends to L entrywise. This in turn implies that, since each Lk is lower triangular with non-negative diagonal entries, L is also.
Read more about this topic: Cholesky Decomposition
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