Positive Definite Matrices
Date: 08/12/2008 at 23:27:25 From: Radhakrishnan Subject: How to determine if a matrix is positive definite? i have a correlation matrix and i want to check if the matrix is a positive definite matrix, if not i want to make the matrix to be positive definite...... in some places I found that a given matrix is positive definite if the Eigen values of the matrix is positive.....i really don't understand what this eigen value is all about or where its applications are? next why should a monte carlo algorithm always wants positive definite covariance matrix? anything on how to approach the problem will be helpfull if you can explain in steps using a small example
Date: 08/15/2008 at 08:21:51 From: Doctor George Subject: Re: How to determine if a matrix is positive definite? Hi Radhakrishnan, Thanks for writing to Doctor Math. There is a lot to your question, like a whole chapter from a linear algebra book. If you can find the book by Gilbert Strang it would help. To determine whether a matrix is positive definite, a standard way is to attempt a Cholesky decomposition. A Google search on "Cholesky decomposition" will give you plenty of information. For your needs you may not need the Eigenvalues. The Cholesky decomposition is useful in monte carlo because it allows you to transform a set of independent normally distributed random variables into a correlated set. It would take a longer discussion to explain what this has to do with positive definiteness. That is where a good text will help. I hope this is enough to get you started. Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/
Date: 08/16/2008 at 04:20:05 From: Radhakrishnan Subject: Thank you (How to determine if a matrix is postive definite? ) Hello George, thank you very much for your answers. i will start studying but at least can you help me with the steps to make a matrix to be positive definite? again, if possible with a small example please.....,.
Date: 08/18/2008 at 08:28:46 From: Doctor George Subject: Re: How to determine if a matrix is postive definite? Hi Radhakrishnan, Correlation and covariance are all about linear relationships between random variables. Let's set up a Monte Carlo simulation of two variables that are normally distributed. The strategy is to start with two independent variables, and then apply a linear transformation to create the desired dependency. Let's call the vector of independent variables X, and the vector of dependent variables Y. Also let L be the matrix that applies the linear transformation. So we have Y = LX If we compute the covariance matrix of Y it will be Cov(Y) = LL^T (L times L transpose) If the rows of L are linearly independent then Cov(Y) will be positive definite. So if we start with Cov(Y), the Cholesky decomposition will find L for us. If the decomposition fails then the matrix was not positive definite, which means that there is no L that will convert X into the desired Y. (I'm not giving a complete proof here, just the strategy. A good text will contain the proof.) Now for the example. Let | 1 0.6 | Cov(Y) = | | | 0.6 1 | The Cholesky decomposition leads to | 1 0 | L = | | | 0.6 0.8 | To perform the simulation, generate values for X1 and X2, and then compute Y1 and Y2. | Y1 | | X1 | | | = L | | | Y2 | | X2 | Does that make sense? - Doctor George, The Math Forum http://mathforum.org/dr.math/
Date: 08/19/2008 at 02:53:07 From: Radhakrishnan Subject: Thank you (How to determine if a matrix is positive definite? ) George, Thank You very much..... it is really helpful. Once again i thank you very much for taking time out of your schedule to help me. As per your advice let me start with a good text. Regards
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