# Questions tagged [cholesky-decomposition]

The Cholesky decomposition is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose.

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### Speeding up Cholesky Decomposition in Gaussian Process Regression

I am currently working on a project that is pulling posterior samples from a Gaussian Process according to the following formula: $Y(x) = \mu(x) + (\Sigma)^{1/2}\mathcal{Z}$ where $\mu(x)$ is a mean ...
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### On the condition of positive semidefiniteness of a matrix

According to Wikipedia, $\textbf{Wikipedia definition:}$ an $n\times n$ Hermitian matrix $M$ is positive semidefinite if and only if it can be decomposed as a product $M = BB^*$ where $*$ is the ...
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### What is the Cholesky decomposition of the sum of a diagonal and a matrix of ones?

What is the Cholesky decomposition $\mathbf{A}=\mathbf{L}\mathbf{L}^\intercal$ of the sum $\mathbf{A}=\mathbf{D}+\mathbf{1}$ of a diagonal matrix $\mathbf{D}$ with only positive elements in the ...
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### Cholesky factorization in L-BFGS-B

L-BFGS-B is one of the most used quasi-Newton solver and the original paper [1] is quite explicit about how to implement the algorithm. In one of the following paper [2], they detail the reference ...
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### Cholesky factorization of $A M A^T$ for $M$ PSD with known Cholesky factorization.

In the context of my research, I am trying to efficiently compute/store a PSD matrix and the cholesky factorization might help. Let $M\in\mathbb R^{n\times n}$ and $A\in\mathbb R^{m\times n}$ be such ...
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### Is the group $Lx+b$ amenable where $L$ is Cholesky?

Let $L$ be any real lower triangular matrix with positive diagonal entries (a Cholesky matrix). Let $x$ and $b$ be real vectors. Is the group of actions $(L, b)$ on $x$, $$L x + b$$ amenable?
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### How can I use this Cholesky decomposition algorithm on this example?

In this course, the authors introduce a method for Cholesky decomposition of matrix $A$, based on row reduction: Procedure 7.4.1: Finding the Cholesky Factorization Using only type 3 elementary row ...
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### Most accurate way to multiply with inverse Cholesky decomposition

What is the most accurate way to compute $x^TA^{-1}y$ for two vectors $x$ and $y$, and a symmetric positive definite matrix $A$? With a Cholesky decomposition $A=LL^T$, one could either apply both $L$ ...
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