# 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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### The Numerical Issues of Computing The QR-Decomposition Using CholeskyQR

I have an application where I need to compute the QR-decomposition (in fact I only need $R$) of a matrix $A\in \mathbb{R}^{m \times n}$ ($m > n$). The matrix has a fixed number of columns, but the ...
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### Let $X,Y,Z$ be three random variables such that the correlation coefficients $\rho_{XY}=0.2, \rho_{YZ}=0.2$, what values can $\rho_{XZ}$ take?

Let $(X,Y,Z)^T$ be jointly normal variable with zero mean such that the correlation coefficients $\rho_{XY}=0.2, \rho_{YZ}=0.2$, what values can $\rho_{XZ}$ take? Prove that there exists a ...
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### Best way to compute $A^{-1}$ when the Cholesky decomposition $A=LL^T$ is known

Suppose $\mathbf{A}$ is symmetric positive definite, and that I have available the Cholesky decomposition of $\mathbf{A}=\mathbf{L}_A\mathbf{L}_A^T$. I want to know $\mathbf{A}^{-1}$. Which of the two ...
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### What is the Cholesky decomposition of an upper triangular matrix?

A similar question is asked here, but for LU decomposition instead of Cholesky, which makes this less trivial. I am using an algorithm (in Matlab) that spends a lot of time computing the Cholesky ...
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### Prove that $\sum=CC^T\Rightarrow \sum^{-1}=C^TC$

For $\sum$ covariance matrix ($n\times n$, symmetric and positive definite) and $C$ lower triangular matrix with real and positive entries, how can I prove the equivalence as from title? I know that ...
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### Reverse software engineering of "\" in Matlab

23.3 Reverse software engineering of "". The following Matlab session records a sequence of tests of the elapsed times for various computations on a workstation manufactured in 1991. For ...
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### Let $A$ be symmetric and positive definite. Suppose $A=LL^T$ is its Cholesky decomposition. Prove that $||A||_2=||L||_2^2$.

Let $A$ be symmetric and positive definite. Suppose $A=LL^T$ is its Cholesky decomposition. Prove that $||A||_2=||L||_2^2$. This is an exercise in my Numerical Analysis book. The offcial hint to this ...
Let $\Sigma$ be a covariance matrix (symmetric positive-definite), and $\Omega = \Sigma^{-1}$ the corresponding precision matrix, which is also SPD (the quotients of positive eigenvalues are positive)....
Let $A=A^T$ be a $n\times n$ positive definite matrix. Define the numbers $f_i(A)=\min\{j:a_{ij}\neq 0\}.$ Consider the following Cholesky decomposition: $A=LL^T$, with $L$ being a lower triangular ...