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Suppose we have a matrix $\mathbf{A}=\begin{bmatrix}a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}$ and know its variance covariance $\left(4\times4\right)$ matrix. Then how the variance covariance matrix of $\mathbf{A}^{-1}$ can be obtained? I would highly appreciate your help. Thanks

MYaseen208
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    What do you call a variance covariance matrix of a matrix ? – Stéphane Laurent Jul 21 '12 at 07:53
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    @StéphaneLaurent I suppose that he meant the covariance of a DATA matrix. In this case, if we label $\vec{a}_1=(a_{11},a_{12})^T$ and $\vec{a}_2=(a_{21},a_{22})^T$, then $\mathbf{A}=(\vec{a}_1,\vec{a}_2)^T$ and the sample variance-covariance matrix the OP's refereing to would have elements $\mathbf{\Sigma}_{ij}=\text{Cov}(\vec{a}_{i},\vec{a_{j}})$. – Néstor Jul 21 '12 at 08:13
  • How can a (data?) matrix 2x2 have covariance matrix 4x4? – ttnphns Jul 21 '12 at 13:22
  • @MYaseen208, ttnphns is right. If it is the covariance of a data matrix, it should be 2x2. – Néstor Jul 21 '12 at 18:50
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    Thanks @Néstor for your comment. It is the variance covariance matrix of the coefficients of the matrix A. – MYaseen208 Jul 21 '12 at 19:14
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    Why don't you just apply the formulae you present at http://stats.stackexchange.com/questions/32696/taylor-series-and-multivariate-delta-method? – whuber Jul 21 '12 at 19:24
  • Thanks @whuber for your comment and suggestion. I doubted about that formula. Do you think it is correct or you can give any reference? Thanks – MYaseen208 Jul 21 '12 at 19:36

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