# 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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### Solving generalized eigenvalue problem with almost positive definitive matrices?

Assume that we are going to solve: $$AV = BVD$$ It's the generalized eigenvalue problem. In this case, $A$ and $B$ are symmetrical. But they are not positive definitive because most of its ...
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### Positive- definiteness of factors of Cholesky factorization

If A is a symmetric positive-definite n×n matrix, then is the lower triangular n×n matrix L positive-definite when A=LL* using Cholesky factorization?
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### Mean orthogonal projection on a vector of multidimensional normal distribution

Let $X\in\mathbb{R}^n$ be a normally distributed random vector with mean zero and positive definite covariance matrix $\Sigma$, i.e. $X\sim\mathcal{N}(0,\Sigma)$. Treating $X$ as a column vector, the ...
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### Apportionate positive and negative value

I have set of values 5,8,10,12 and i have another adjustment value 20. i wanted adjustment has to be apportioned to other set of values. I did like 5+8+10+12=35. 5/35*20 for 5 and likewise for rest ...
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### How to quickly determine the definiteness of a large sparse matrix without using Sylvester's criterion?

I am currently trying to classify the stationary points of a function as either a maximum, minimum or saddle points based on the definiteness of the Hessian at those points. I have worked out that ...
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### proving that lu decomposition is not unique on singular matrix.

How to prove that the following isn't true (using 3 by 3 matrix): Given A is a square and a singular matrix (which means non invertible), if LU decomposition is possible without the use of ...
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In some textbooks it is mentioned that, if a matrix is Positive Semidefinite, then all its diagonal entries are nonnegative and all the principal submatrices of PSD obtained by removing any number of ...
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### Does positive definiteness imply $x^{T}Mx \geq \alpha x^{T}x$ for some $\alpha > 0$?

I require an $n \times n$ matrix to satisfy $x^{T}Mx \geq \alpha x^{T}x$ for some $\alpha > 0$ for all $x \in \mathbb{R}^{n}$ where $x \geq 0$ (1). Is it sufficient to show that M is positive ...
I have read in the post, but there is something missing I'd like to know. I do believe it's not a duplicate post. I have a specified question, probably a trivial one. First, let $(s_n)_{n\geq 0}$ be ...