Hi. I have a real tridiagonal symmetric matrix that comes from the discretization of a partial differential equation. The elements are given by:(adsbygoogle = window.adsbygoogle || []).push({});

##A_{i,j}=-\delta_{i-1,j}\kappa_{i-1/2,l}\frac{\Delta t}{h^2}+\delta_{i,j}\left[\frac{2}{c}+\frac{\Delta t}{2}\mu_{i,j}+\frac{\Delta t}{h^2}\left(\kappa_{i+1/2,l}+\kappa_{i-1/2,l}\right) \right]-\delta_{i+1,j}\kappa_{i+1/2,l}\frac{\Delta t}{h^2}##.

##\kappa## and ##\mu## are discretized functions, and both are positive (##\mu## might be zero at some points, but never negative, and ##\kappa## is strictly positive). This is a symmetric tridiagonal matrix. I would like to know if it is positive definite.

Now I have read somewhere (I don't know where, but I took note) that if the product of the elements of the matrix:

(1) ##A_{i,i+1}A_{i+1,i}>0##,

then all eigenvalues of ##A##, let's say ##\lambda_n>0##, and therefore ##A## is positive definite, this is accomplished for my matrix. So is ##A## positive definite? I think it is under the assumption (1) I've made, but I don't know where the theorem that gives condition (1) and ensures that the eigenvalues are positive comes from.

Thanks in advance.

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# I How do I know if a matrix is positive definite?

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