Describe the mistake
In Theorem 4.18, The last sentence is 'L is unique'.
Location
Please provide the
version (bottom of page) - Draft (2021-07-29)
Chapter - 4.3 Cholesky Decomposition
page - p114
line number/equation number - The last line of Theorem 4.18
Proposed solution
I think it should be revised like 'L is unique if diagonal elements are restricted to be positive.'
Assume that A = LL^T = MM^T.
I = L^(-1)MM^(T)L^(-T) <=> (L^(-1)M)^(-1) = (L^(-1)M)^T
L^(-1) is upper triangular matrix and M is lower triangular matrix based on the definition of Cholesky Decomposition.
So, L^(-1)M is a diagonal matrix.
D := L^(-1)M <=> D^2 = I
Therefore, D = I or -I
It means that we have 2 Cholesky factors (+L and -L).
Additional context
Add any other context about the problem here.
Describe the mistake In Theorem 4.18, The last sentence is 'L is unique'.
Location Please provide the
Proposed solution I think it should be revised like 'L is unique if diagonal elements are restricted to be positive.'
Assume that A = LL^T = MM^T. I = L^(-1)MM^(T)L^(-T) <=> (L^(-1)M)^(-1) = (L^(-1)M)^T L^(-1) is upper triangular matrix and M is lower triangular matrix based on the definition of Cholesky Decomposition. So, L^(-1)M is a diagonal matrix. D := L^(-1)M <=> D^2 = I Therefore, D = I or -I It means that we have 2 Cholesky factors (+L and -L).
Additional context Add any other context about the problem here.