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I try to understand and visualize myself covariance matrix. Supposing I have a matrix A = [ 2 3 4; 5 5 6 ], how do I calculate its covariance matrix, and what is its practical meaning? (All I was able to understand by now is that on the diagonal of the covariance matrix, variances for particular variables are placed, and on the upper and lower fields correlations between those variables.)

josh130
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  • Please also see concise an comprehensive explanation of covariance and related matrices via matrix multiplication. – ttnphns Apr 16 '13 at 11:58
  • As far as I know, a covariance matrix can be computed for (a) multivariate distributions and (b) multivariate samples, but not for arbitrary $m$ by $n$ matrices such as your A, so it's impossible to tell what you're asking. Could you clarify? – whuber Apr 16 '13 at 15:31

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The off diagonal entries are typically covariances rather than correlations. The practical meaning depends on the context. In the most simple case it is just a table summarizing the variation within variables and the strength of the bivariate linear association between variables. For some models (e.g. SEM or LISREL models) the covariance matrix contains all the information necessary to estimate it.

Maarten Buis
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If you have a vector of random variables $X = (X_1,...,X_n)^T$, the co-variance matrix $\Sigma$ can be calculated finding the pairwise co-variance of each of the variables. E.g. $\Sigma_{ij} = Cov(X_i,X_j)$.

So, like you say, along the diagonals you have $\Sigma_{ii} = Cov(X_i, X_i)=Var(X_i)$ and the upper and lower have the co-variance for each pair of variables. The co-variance is numerically describing how each of the variables vary with each other.

I am not sure what you mean by finding the co-variance of a matrix. Perhaps you desire the co-variance matrix between two column vectors?

jatotterdell
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