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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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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Inertia of reduced symmetric real regular indefinite matrix
Given is a regular real symmetric matrix $\boldsymbol{A} \in \mathbb{R}^{n \times n}$, with $m < n/2$ negative eigenvalues.
I wish to know: Is it always possible to extract a submatrix of $k=n-2\...
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Gradient with respect to $LDL^\prime$ parameterization of covariance matrix
I have been working with the matrix-variate normal distribution (a.k.a., matrix normal distribution) $\mathbf{X} \sim \text{Normal}_{nm}\big(\mathbf{M},\;\mathbf{I}_n,\mathbf{V}\big)$, such that (...
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Endogeneity Analysis without the access of raw data?
I currently have the correlation/covariance matrix for a set of variables and the results of regression analysis but lack access to the raw dataset. Under these constraints, is it feasible to conduct ...
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Factorizing $AMA^T+N=WW^T$ efficiently.
This question is related to this question with a slightly more general setting. A good speedup on this could improve performances of a decision process in multi-objective optimization that I designed.
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Random bounded triangular $T_n$ s.t. $Var[T_nS_nT_n^\top]\to 0$ for nonrandom psd $S_n$. Does $(T_n-\bar T_n)S_n\to^P0$ for some nonradom $\bar T_n$?
Let $T_n$ be a sequence of square random matrices with $T_n$ lower triangular with $diag(T_n)=(1,1...,1)$ and $S_n$ a sequence of deterministic symmetric psd matrices. All matrices are in $R^{d\times ...
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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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Prime numbers and a positive definite matrix?
Probably this has nothing to do with prime numbers, I just experimented a little bit with it and wanted to share it, in case someone has an idea.
Let
$$p_n := n\text{-th prime number , }[a,b]:= \frac{...
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SVD or Cholesky on sum of SPD matrices
Let $A$ and $B$ be symmetric positive definite (SPD) matrices and $C=A+B$. I know the SVD or Cholesky decomposition of A and B, $A=U_A\Sigma_AU_A^T=L_AL_A^T$ and $B=U_B\Sigma_BU_B^T=L_BL_B^T$.
Can I ...
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Which row/column to remove from SPD matrix to remain maximal volume
Let $A$ be a real $N\times N$ symmetric, positive definite (SPD) matrix with volume $vol(A)=|det(A)|$. Let $B_i$ be the matrix $A$ where row and column $i$ were exchanged by a unit vector $e_i$. Can I ...
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Equivalence of the LDL decomposition with an upper-triangular or lower-triangular matrix
I am aware that given a positive-definite matrix $A$ we can compute its LDL decomposition as:
$$ A = L D L^t $$
where $L$ is a lower unit triangular matrix and $D$ a diagonal matrix.
In this paper by (...
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