# Questions tagged [positive-definite]

For questions about positive definite real or complex matrices. For questions about positive semi-definite matrices, use the (positive-semidefinite) tag.

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### Is there any hint how to prove this?

Let's consider matrix M is defined as follows: $M = \begin{bmatrix} P & v \\ v^t & d \end{bmatrix}$, where $P \succ 0$, d is a scalar, and v is a vector. Problem: :To $M \succ 0$ be a ...
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### Question about positive definite matrix and inequality proof

Problem： Let $X$ and $R$ be positive definite matrices, $C$ is a matrix of compatible dimension, and define $g(X)$ as $g(X)=X-XC'[CXC'+R]^{-1}CX$ Prove that if $X>Y>0$, then $g(X)>g(Y)$. From ...
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### How to show that all of the all eigen values of the matrix are positive?

I have a matrix \begin{bmatrix} 6I &-4I&0&0&......0\\ -4I &6I& -4I & 0&......0\\ 0 & -4I & 6I & -4I &......0\\ .\\ .\\ .\\ 0&0&0&........-4I&...
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### What functions preserve symmetry and positive-definiteness of covariance matrices?

Suppose I have covariance matrices $H_1,…,H_n$ (symmetrical, positive-definitive) and corresponding weights $w_1,..,w_n$. We want to find a function $f$, such that $H = f(H_1,…,H_n,w_1,…,w_n)$ is ...
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### Solving for a Diagonal Positive Definite Matrix: $\Delta$ such that $b'\Delta C'(C\Delta C')^{-1}=a'DC'(CDC')^{-1}$

Any help with the following conjecture would be highly appreciated. I have a hard time figuring out how to start. Though counterexamples would be certainly helpful, if the conjecture is not always ...
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### Decomposition of positive definite matrix.

Let $A$ be a positive definite matrix. Following are the decomposition's of matrix $A$: $A = PP$ $A = MM^{T}$ Empirically we found that $||P||_{F}^2 = ||M||_{F}^2$ but how to prove this ...
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### Inverse of a positive symmetric matrix with large entries

I have a random vector $Z = X_{1} + X_{2} + \ldots + X_{T}$, where each $X_{i}$ is a $n\times 1$ random vector with finite mean and covariance matrix. Each $X_i$ has a different mean and covariance ...
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### Deriving a Cholesky decomposition algorithm

I am studying numerical linear algebra using Lloyd Trefethen's book. In the chapter on systems of linear equations, I am reading about the Cholesky decomposition. I understand existence and uniqueness,...
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### Uniform sampling on the set of symmetric positive-semidefinite matrices with bounded entries

What is the most correct way to randomly generate a (square) symmetric positive-definite matrix $A$ with nonnegative entries bounded in [0,1]? One way I can think of is by sampling a matrix $X$ from ...
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### Proof of "If M and N are positive definite, then the products MNM and NMN are also positive definite"

Here on Wikipedia, it states that: "If M and N are positive definite, then the products MNM and NMN are also positive definite" I've tried looking for a proof of this statement, but I cannot ...
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### How does one denote the set of all positive-definite square matrices? [duplicate]

For example, can I write: The matrix $X \in \mathbb{R}^{p \times p}_{>0}$ follows a Wishart distribution $$X \sim \mathcal{W}(V,n)$$ where $\mathbb{R}^{p \times p}_{>0}$ is the set of all ...