On Wishart Distributions

Asked Oct 06 '15 at 02:32

Active Apr 27 '22 at 11:01

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Notation: Let $S$ is a complex square matrix. ${S}^H$ denotes conjugate transpose matrix, while $[S]_{k,k}$ denotes the k-th diagonal element, and $S^{-1}$ denotes the inverse matrix.

Let $S=GG^{H}$ be a positive definite matrix where all elements of $G$ are independent complex Gaussian distributed. $S$ is said to have a Wishart distribution, written as $S\sim\mathcal{W}_p(n,\Sigma)$ with $p<n$.

My concern: What are the distributions of $[S]_{k,k}$ and $[S^{-1}]_{k,k}$? Although these distributions arise very frenquently in applied science (e.g., see [R1]), it does not seems to have proper novel available. I am really interested in such distributions. Can anyone help me, please?

R1: Transmit Selection in Spatial Multiplexing Systems

edited Apr 27 '22 at 11:01

Glorfindel

3,955

asked Oct 06 '15 at 02:32

pej

1

The elements of $G$ are independent standard normal, and a diagonal element $[S]_{k,k}$ is the sum of square of k'th row, i.e. sum of squares of i.i.d standard normals, hence chi-squared distributions with k degree of freedom. – Weaam Oct 06 '15 at 02:56
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More generally, check "Distribution of Diagonal Elements of a General Central Complex Wishart Matrix", IEEE Communications Letters, V.16 N.9, 2012. And for diagonal of inverse, see the post Marginal distribution of the diagonal of an inverse Wishart distributed matrix – Weaam Oct 06 '15 at 03:09

On Wishart Distributions

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